Income Insurance and the Equilibrium Term-Structure of Equity · Income Insurance and the...

84

Transcript of Income Insurance and the Equilibrium Term-Structure of Equity · Income Insurance and the...

Income Insurance and the Equilibrium Term-Structure

of Equity

ROBERTO MARFE∗

Journal of Finance, forthcoming

ABSTRACT

Output, wages and dividends feature term-structures of variance-ratios respectively flat,

increasing and decreasing. Income insurance from shareholders to workers empirically

and theoretically explains these term-structures. Risk sharing smooths wages but only

concerns transitory risk and, hence, enhances the short-run dividend risk. A simple

general equilibrium model, where labor rigidity affects dividend dynamics and the price

of short-run risk, reconciles standard asset pricing facts with the term-structures of

equity premium and volatility and those of macroeconomic variables, at odds in leading

models. Consistently, actual labor-share variation largely forecasts dividend strips risk,

premium and slope.

JEL Classification: D51, E21, G12

∗Collegio Carlo Alberto. Email: [email protected]. I am grateful to Kenneth Singleton(the Editor), an Associate Editor and two referees for their comments and suggestions. I would like tothank my supervisor Michael Rockinger and the members of my Ph.D. committee (Swiss Finance Institute)Philippe Bacchetta, Pierre Collin-Dufresne, Bernard Dumas, and Lukas Schmid for stimulating conversationsand many insightful comments and advice. I would also like to acknowledge comments from Hengjie Ai, DanielAndrei, Marianne Andries (Stockholm discussant), Ainhoa Aparicio-Fenoll, Ravi Bansal, Alessandro Barattieri,Cristian Bartolucci, Luca Benzoni, Jonathan Berk, Harjoat Bhamra, Vincent Bogousslavsky, Maxime Bonelli,Andrea Buraschi, Gabriele Camera, Gian Luca Clementi (EFA discussant), Stefano Colonnello, GiulianoCuratola, Marco Della Seta, Jerome Detemple (Gerzensee and Paris discussant), Theodoros Diasakos, JackFavilukis, Jordi Galı, Christian Gollier, Edoardo Grillo, Campbell Harvey, Michael Hasler, Marit Hinnosaar,Toomas Hinnosaar, Christian Julliard, Leonid Kogan, Kai Li, Jeremy Lise, Elisa Luciano, Sydney Ludvigson,Loriano Mancini, Charles Manski, Guido Menzio, Antonio Merlo, Ignacio Monzon, Giovanna Nicodano, MiguelPalacios (Rome discussant), Francisco Palomino, Julien Penasse, Ilaria Piatti, Andrea Prat, Dan Quint, JeanCharles Rochet, Birgit Rudloff, Yuliy Sannikov, Jantje Sonksen, Raman Uppal, Ernesto Villanueva and fromthe participants at the 2nd BI-SHoF Conference in Asset Pricing and Financial Econometrics, at the CSEF-EIEF-SITE Finance & Labor Conference 2015, at the 32nd International Conference of the French FinanceAssociation 2015, at the 18th Annual Conference of the Swiss Society for Financial Market Research 2015(SGF), at the 41st European Finance Association Annual Meeting 2014 (EFA), at the 11th International ParisFinance Meeting 2013 (AFFI / EUROFIDAI), at the 12th Swiss Doctoral Workshop in Finance 2013, and atseminars at the Collegio Carlo Alberto, Essec Business School, Laval University. All errors remain my onlyresponsibility. Part of this research was written when the author was a Ph.D. student at the Swiss FinanceInstitute and at the University of Lausanne and a visiting scholar at Duke University. Financial support bythe NCCR FINRISK of the Swiss National Science Foundation and by the Associazione per la Facolta diEconomia dell’ Universita di Torino is gratefully acknowledged. Usual disclaimer applies. The author has noconflict of interests and has nothing to disclose.

Leading asset pricing models describe many characteristics of financial markets but fail to

explain the timing of equity risk. These models have different rationale (e.g. habit formation,

time-varying expected growth, disasters, prospect theory) but share an important feature:

priced risk comes from variation in long-run discounted cash-flows.1 Instead, van Binsbergen,

Brandt, and Koijen (2012), van Binsbergen, Hueskes, Koijen, and Vrugt (2013) and van

Binsbergen and Koijen (2016) document that the term-structures of equity volatility and

premia are downward sloping –that is markets compensate short-run risk. Moreover, standard

models usually base on assumptions concerning dividend dynamics which imply an upward-

sloping term-structure of dividend risk (i.e. volatilities or variance-ratios of dividends’ growth

rates which increase with the horizon). Instead, consistently with Belo, Collin-Dufresne, and

Goldstein (2015), this paper documents that dividend risk is strongly downward-sloping, such

that many models overestimate long-horizon dividend risk by an order of magnitude.

Why is dividend risk downward-sloping? Under which conditions, does downward-sloping

dividend risk transmit to equity risk and premia? What macroeconomic channel explains

short-term equity returns? This paper empirically and theoretically addresses these questions

by providing a macroeconomic foundation of the timing of risk and by reconciling, in equilib-

rium, standard asset pricing facts with the new evidence about the term-structures. This is

important because the term-structures of both fundamentals and equity provide information

about how prices are determined in equilibrium. Hence, a term-structure perspective offers

additional testable implications to asset pricing frameworks and can help us to understand

the macroeconomic determinants of asset prices.

The paper argues that labor rigidity is at the heart of the timing of macroeconomic risk.

Danthine and Donaldson (1992, 2002), among others, show that a mechanism of income in-

surance from shareholders to workers, which takes place within the firm, leads to volatile and

pro-cyclical dividends. Hence, this mechanism explains why equity commands a high compen-

sation.2 Beyond such a cyclicality effect of income insurance, I show that also a term-structure

effect takes place. Since output, wages and dividends are co-integrated (Lettau and Ludvig-

son, 2005), income insurance implies that the transitory component of aggregate risk is shared

1

asymmetrically among workers and shareholders, whereas the permanent component is faced

by both. Namely, wages are partially insured with respect to transitory risk, whereas divi-

dends load more on transitory risk as a result of operating leverage. Thus, wage and dividend

risks shift respectively toward the long and the short horizon and workers and shareholders

bear respectively more long-run risk and more short-run risk. I embed this mechanism of

income insurance in an otherwise standard closed-form general equilibrium model and show

that, under standard preferences, the term-structure effect of income insurance is inherited by

financial markets and leads to downward sloping term-structures of equity risk and premia.

Consistently, actual labor-share variation largely predicts short-term equity risk, premium and

slope.

An empirical investigation supports the main model mechanism. First, I document that

the timing of risk of macroeconomic variables is heterogeneous. The term-structures of risk

of output, wages and dividends are respectively flat, increasing and decreasing. In accord

with the model, these term-structures –and the co-integrating relationship among the lev-

els of these variables– support the idea that both workers and shareholders are subject to

permanent shocks but they share transitory shocks asymmetrically, as a result of income in-

surance. Consistently, Gamber (1988) and Menzio (2005) show that implicit contracts and

labor market search frictions lead to wage rigidity over transitory risk only; Guiso, Pistaferri,

and Schivardi (2005) and Ellul, Pagano, and Schivardi (2014) provide evidence that insurance

does not concern permanent shocks; and Rıos-Rull and Santaeulalia-Llopis (2010) document

that the wage-share is stationary and counter-cyclical.

Second, I further support the term-structure effect of income insurance by investigating

the model prediction that the variance ratios of dividends and the gap between the variance

ratios of wages and dividends should be respectively decreasing and increasing with the labor-

share. The model predicts that, after a negative transitory shock, wages are partially insured

whereas dividends are hit more. This implies that, on the one hand, wages are high relative

to dividends (the cyclicality effect) and that wages load less than dividends on transitory

risk (the term-structure effect). Therefore, the distance between the upward-sloping variance

2

ratios of wages and the downward-sloping variance ratios of dividends should increase when

the labor-share is high. The opposite holds after a positive transitory shock. I provide robust

empirical evidence that such a dynamic relation obtains in the actual data.

Third, in accord with the model, I document an extremely strong connection between labor-

share variation and short-term equity returns. The correlations between the labor-share and

one-year ahead dividend strips volatility (risk), excess return over the risk-free rate (premium)

and excess return over the market return (slope) are respectively 86%, 57% and 52%.

Moreover, I show that the remainder of output minus wages features a term-structure

of risk which is markedly downward-sloping and essentially recovers the negative slope of

dividend risk. This implies that the bulk of the gap between the approximatively flat variance-

ratios of output and the decreasing variance-ratios of dividends should be imputed to labor.

Consistently, I document that the transitory component of dividends that should not be

imputed to income insurance does not help to explain either the negative slope of dividend

risk or dividend strip returns. Finally, the term-structure effect of income insurance implies

that the labor-share positively forecasts dividend and consumption growth, which I show to

be a robust feature of actual data.

The model consists of a few simple ingredients and is similar to Greenwald, Lettau, and

Ludvigson (2014). Investment is not directly modelled and total resources are shared by

workers and shareholders. The former do not invest and consume their wages (Berk and

Walden, 2013), and the latter receive dividends and, thus, act as a representative agent on the

financial markets. Total resources are subject to both permanent and transitory technological

shocks (Bansal, Kiku, and Yaron, 2010; Lettau and Ludvigson, 2014).3 Income insurance

concerns transitory risk, as suggested by the empirical analysis. The insurance mechanism, on

the one hand, generates counter-cyclical labor-share and downward-sloping dividend risk and,

on the other hand, enhances the pricing of short-run risk. Indeed, the equilibrium state-price

density equals the shareholders’ marginal utility but is affected by income insurance through

the levered dynamics of dividends. Differently from Danthine and Donaldson (2002), the

model leads to a stationary equilibrium even in presence of permanent shocks (time-varying

3

long-run growth as in Bansal and Yaron (2004)) and recursive preferences. Under standard

preferences (i.e. intertemporal substitution larger than wealth effect and preference for the

early resolution of uncertainty), the term-structure of equity premia is non-monotone: long-

run growth leads to an upward-sloping effect and transitory risk to a downward-sloping one.

Indeed, transitory shocks are expected to recover and, hence, are risky in the short-run and

safe in the long-run. For income insurance strong enough, operating leverage enhances the

transitory risk of dividends and its equilibrium price. In turn, the negative slope of dividend

risk transmits to equity returns.

The model calibration exploits the information from the term-structures of aggregate con-

sumption, wage and dividend risk. On the one hand, they represent additional stylized facts

captured by the model; on the other hand, the timing of macroeconomic risk allows to quan-

titatively measure the strength of income insurance and, hence, the operating leverage on

dividends. Moreover, the term-structures of risk provide information about the persistence

of latent factors –a main issue in asset pricing models. The baseline calibration reconciles

the term-structures of both fundamentals and equity with the traditional asset pricing facts,

such as the risk-free rate and equity premium puzzles as well as the the excess-volatility of

equity (also at long-horizons, as in Beeler and Campbell (2012)). In accord with the data, the

slope of equity premia is due to both the cash-flows and the discount rates channels. More-

over, the model leads to upward-sloping real interest rates. van Binsbergen et al. (2013) and

Aıt-Sahalia, Karaman, and Mancini (2015) refine the findings of van Binsbergen et al. (2012)

suggesting that the negative slope of equity premia is an unconditional property of the data,

whereas conditional equity premia are slightly increasing in good times and strongly decreasing

in bad times. Such dynamics obtain in the model –without deteriorating other results– once

fundamentals feature business cycle uncertainty. Time-varying risk premia lead to a pattern

of long-horizon predictability consistent with actual data. Most of general equilibrium asset

pricing models –such as the long-run risk model of Bansal and Yaron (2004), which obtains

as a sub-case– cannot explain simultaneously all of these stylized facts. Closed-form solutions

allow to decompose the equity premium and to derive an equilibrium density which captures

4

the relative contribution of discounted cash-flows at each horizon. In the baseline calibration,

the horizon that mostly contributes to the equity premium is the immediate future, that is the

density is monotone decreasing (whereas it is hump-shaped in a long-run risk model). This

suggests that a “short-run” explanation of financial markets is needed to reconcile the timing

of macroeconomic risk with the traditional asset pricing facts.

Related literature. A few recent papers suggest alternative and complementary channels

which can contribute to explain the term-structure of equity. See van Binsbergen and Koijen

(2016) for a recent review.4

The economic mechanism in Belo et al. (2015) differs from that of this paper: they argue

that the negative slope of dividend risk comes out from rigidity in financial leverage. Namely,

financial leverage switches upward-sloping EBIT risk to downward-sloping dividend risk. How-

ever, I empirically show that the term-structure of risk of EBIT is markedly downward-sloping

and close to that of dividends. Thus, the financial leverage channel can only have a limited

term-structure effect. Most of the change in the slope obtains earlier than financing decisions

and should be imputed to wages. Croce, Lettau, and Ludvigson (2015) study the investors’ be-

liefs about fundamentals. Namely, they assume the usual long-run risk dynamics for dividends

and show that bounded rationality under limited information can lead to downward-sloping

dividend risk under the subjective measure of the investors. While this mechanism is comple-

mentary to that of this paper, Croce et al. (2015) assume that dividend risk is upward-sloping

under the physical measure. Instead, dividend risk is markedly downward-sloping in actual

data. Then, the beliefs formation channel can only have a limited term-structure effect rel-

ative to the bulk of the term-structure effect driven by wages. Ai, Croce, Diercks, and Li

(2015) propose a general equilibrium model where heterogeneous investment risk due to vin-

tage capital leads to a downward-sloping term-structure of equity premia. The investment

channel is potentially complementary to the income insurance mechanism. However, on the

one hand, the term-structure of risk of output minus wages almost recovers the negative slope

of dividend risk; on the other hand, the term-structure of risk of output minus investments

is almost identical to that of output. Thus, investments do not induce a downward-sloping

5

effect. Both Belo et al. (2015) and Kogan and Papanikolaou (2015) are partial equilibrium

models: given dividends, a downward-sloping term-structure of equity premia obtains because

of the exogenous correlations between the priced factors and dividend risk in those economies.

A general equilibrium approach would produce economic restrictions on those correlations and

offer additional insights into the relationship between either financial leverage or investment-

specific risk and equity term-structures. Hasler and Marfe (2015) suggest that the negative

slope of dividend risk is due to post-disaster recovery. As long as dividends recover after being

hit by a disaster, dividend risk shifts towards the short horizon. While this mechanism is both

alternative and complementary to that of income insurance, they interact each other since

usually human capital is affected by disasters to a lower extent than physical capital.

A number of recent papers points out the importance of labor relations for asset pricing.

This paper complements such a literature investigating the term-structure implications of in-

come insurance in general equilibrium for both dividends and equity. Note that labor markets

frictions –such as infrequent wage resettling in Favilukis and Lin (2015), search frictions in

Kuehn, Petrosky-Nadeau, and Zhang (2012) and labor mobility in Donangelo (2014)– can

lead to a similar term-structure effect. I model income insurance as an empirically motivated

risk-sharing rule to capture in reduced form the combinations of labor markets mechanisms

and frictions which give rise to labor rigidity. While Danthine and Donaldson (1992, 2002),

Favilukis and Lin (2015) and Kuehn et al. (2012) propose big macro-finance models and rely

on numerical solutions, I consider a simpler and more parsimonious economy and provide ana-

lytical solutions.5 Recently, Greenwald et al. (2014) investigate the asset pricing implications

of low frequency swings in labor-share.

The model has also implications for the cross-section of equity returns. Lettau and Wachter

(2007, 2011) show in partial equilibrium that a model which captures the downward-sloping

term-structure of equity automatically gives rise to a value premium, when value firms are

defined as shorter duration equity than growth firms. The term-structure effect of income

insurance provides a foundation to the intuition of Lettau and Wachter (2007). Indeed, Marfe

(2015a) shows that such a result obtains in an extended version of the model and provides

6

empirical support documenting a strong relation between variation in the labor-share and the

value premium dynamics. Thus, the model is also consistent with Koijen, Lusting, and van

Nieuwerburgh (2014), who empirically document that markets compensate short run-risk and

that the value premium dynamics is related to business cycle risk.

The model emphasis on the fundamentals transitory risk is key to understand the term-

structure of equity. However, the model has some limitations in the description of real interest

rates. Namely, the model qualitatively captures the positive sign of the slope and the long-run

real yield, but it quantitatively overestimates the curve level.6 This can be explained by the

extreme form of limited market participation assumed in the model. Since workers do not

access financial markets, the state price density inherits the large transitory risk of dividends,

and bonds are essentially hedge instruments against dividend risk. This appears in contrast

with the results by Alvarez and Jermann (2005), who infer a prominent role of permanent risk

in discount rates.7 Such a tension of the model can be eventually relaxed in a framework that

endogenizes participation in stock and bond markets, in spirit of Berk and Walden (2013).

The paper is organized as follows. Section I provides empirical support to the main model

assumptions, mechanism and predictions. Section II describes the economy and derives the

equilibrium asset pricing predictions. Asset pricing results are in Section III. A model exten-

sion is discussed in Section IV. Section V concludes.

I. Empirical Analysis

The first part of this section documents that the timing of risk is heterogeneous among macroe-

conomic variables. The second part argues and empirically supports the idea that a mechanism

of income insurance between workers and shareholders is at the heart of the timing of risk.

The third part shows that income insurance is a main driver of dividend strips risk and pre-

mia. The final part provides further robustness. The remainder of the paper shows that an

otherwise standard general equilibrium model with income insurance reconciles standard asset

pricing facts with the new evidence about the term-structure of equity.

7

I consider postwar data from the U.S. non-financial corporate sector (such as value added,

labour compensations, EBIT and dividends from the Flow of Funds).

A. The Timing of Macroeconomic Risk

I define the timing of risk of a given variable as the term-structure of its variance ratios (VR’s

hereafter, i.e. the ratio of the growth rates variance at horizon τ relative to τ times the

variance at horizon one). VR’s capture whether the variance of a given variable increases

linearly with the observation interval. Hence, downward-sloping VR’s below unity imply that

risk concentrates at short horizons. Vice-versa, upward-sloping VR’s above unity imply that

risk concentrates at long horizons.

The left panel of Figure 1 provides a number of insights. We observe that the VR’s of

wages are markedly upward-sloping and above unity, whereas the VR’s of EBIT and dividends

are markedly downward-sloping and below unity. Instead, the VR’s of value added lie in the

middle and are approximatively flat. These results are consistent with and extend those

in Beeler and Campbell (2012) and Belo et al. (2015). Note that the usual assumptions

of asset pricing models about the dynamics of dividends are strongly inconsistent with the

data. Indeed, time-variation in the conditional moments of a geometric process (e.g. long-run

growth, stochastic volatility, time-varying disasters) induce markedly upward-sloping VR’s.

Thus, long-horizon dividend risk is usually overestimated by an order of magnitude or more.8

Figure1

B. Does Income Insurance Explain the Timing of Risk?

What explains the heterogeneity in the timing of macroeconomic risk? I argue that the main

driver is an explicit or implicit mechanism of income insurance from shareholders to workers

that takes place within the firm. The reasoning is the following.

On the one hand, Danthine and Donaldson (2002) among others suggest that risk-sharing

between workers and shareholders (as well as labor markets frictions) leads to smooth wages

8

and, as a result of operating leverage, risky dividends. Hence, the labor-share is counter-

cyclical and the dividend-share is pro-cyclical, as documented by Rıos-Rull and Santaeulalia-

Llopis (2010). I name this mechanism as the cyclicality effect of income insurance.

On the other hand, Guiso et al. (2005) document that such an income insurance exists

but does not concern permanent shocks. Menzio (2005) theoretically shows and empirically

supports the idea that labor markets frictions lead to wage rigidity with respect to transitory

shocks instead of permanent shocks. Consistently, Lettau and Ludvigson (2014) document that

macroeconomic variables are subject to both permanent and transitory shocks and Lettau and

Ludvigson (2005) document that wages and dividends are co-integrated: i.e. they face the

same permanent shocks. I extend this result and provide evidence that all quantities in Figure

1 are co-integrated (see the Online Appendix A).

Therefore, beyond the cyclicality effect of income insurance, also a term-structure effect

takes place. Given co-integration, the mechanism of income insurance only concerns transi-

tory shocks. Thus, after a negative transitory shock, the fraction of total resources devoted to

workers’ remuneration is partially insured by a decrease of the resources devoted to sharehold-

ers’ remuneration. Such an insurance mechanism implies that the labor- and dividend-shares

are respectively decreasing and increasing in the transitory component of output. As a conse-

quence, wages and dividends load on transitory risk respectively less and more than output.

This produces VR’s of wages and dividends which are respectively increasing and decreasing

and respectively above and below the VR’s of output. Such a term-structure effect of income

insurance coincides with the empirical VR’s in the left panel of Figure 1.

The right panel of Figure 1 provides in a intuitive way strong empirical support to the term-

structure effect of income insurance and its magnitude. Indeed, the VR’s of the remainder

of value added minus wages recover the negative slope of the term-structures of both EBIT

and dividends. This means that wages explain the bulk of the distance between the flat term-

structure of value added and the downward-sloping term-structure of dividends. Namely,

increasing and decreasing VR’s of wages and dividends represent the other side of the same

coin of respectively wage rigidity and operational leverage on dividends. Instead, alternative

9

channels such as investment (suggested by Ai et al. (2015) and Kogan and Papanikolaou

(2015)) or financial leverage (suggested by Belo et al. (2015)) have a little or negligible

impact on the timing of dividend risk.9 In the following, I provide more formal empirical

support to the cyclicality and term-structure effects of income insurance.

B.1. The Cyclicality Effect of Income Insurance

Accordingly with the above reasoning, the cyclicality effect of income insurance should imply

that: i) changes in the labor-share are negatively correlated with changes in output; and ii)

the labor-share and the dividend-share are negatively correlated.

Consistently with a number of previous works, I find empirical support for both these

stylized facts. The labor-share is counter-cyclical –i.e. the correlation of changes in the labor-

share and changes in log value added is -40.2%– and moves negatively with the dividend-

share –i.e. the correlation between the labor- and dividend- shares is -23.5%. These results

resemble the empirical findings of Boldrin and Horvath (1995), Shimer (2005) and Rıos-Rull

and Santaeulalia-Llopis (2010). The Online Appendix A documents that the labor-share

Granger causes the dividend-share, whereas investment and financing decisions do not.

The smoothing effect of income insurance on wages can be easily observed. The negative

covariance between labor-share and value added more than offset the contribution of the

labor-share variation to wage variance:

1 = var(∆ log Y )var(∆ logW )

+ var(∆ logW/Y )var(∆ logW )

+ 2 cov(∆ log Y,∆ logW/Y )var(∆ logW )

= 149.1% + 48.4% − 97.5%.

Dynamically we should observe a decrease in wage volatility and an increase in dividends

volatility when the labor-share is high. Then, I regress a moving-average of growth rates

10

conditional volatility on a moving-average of the labor-share in post-war quarterly data:

σWt,t+1 = a − 0.036(NW-t: -2.02)

W/Y t + εt, adj-R2 = 9.32%,

σDt,t+1 = a + 0.689(NW-t: 4.15)

W/Y t + εt, adj-R2 = 13.61%.

Coefficients are statistically significant and their sign is consistent with income insurance.10

B.2. The Term-Structure Effect of Income Insurance

If income insurance only concerns transitory shocks, wages and dividends load respectively less

and more than output on transitory risk. Given co-integration, this leads to a term-structure

effect. Namely, after a negative (positive) transitory shock, workers’ remuneration increases

(decreases) relative to total resources, and the VR’s of wages and dividends respectively in-

crease and decrease (decrease and increase) relative to those of output.

To test this dynamic relation, I verify whether time-variation of VR’s is consistent with

the mechanism of income insurance. Namely, I build time-series of VR’s of wages, dividends

and output. Then, I regress the VR’s of wages, the VR’s of dividends and their difference on

the labor-share:

VRW

∣∣t,τ

= a + bwWY t

+ byVRY

∣∣t,τ

+ εt,

VRD

∣∣t,τ

= a + bdWY t

+ byVRY

∣∣t,τ

+ εt,

VRW − VRD

∣∣t,τ

= a + bwdWY t

+ byVRY

∣∣t,τ

+ εt,

for several horizons τ ranging from 2 to 7 years. I include the VR’s of output as a control

to account for the variation in the VR’s due to permanent shocks. Accordingly with the

above reasoning, coefficients bw and bwd should be positive and the coefficient bd should be

negative. I perform these regressions using quarterly data on the sample 1951:4-2015:1 from

the non-financial corporate sector. Estimation results are reported in Table I. TableI

Sign and significance of coefficients strongly support the term-structure effect of income

11

insurance. When resources devoted to workers’ remuneration are high (low) relative to total

resources, dividends load more (less) on transitory risk and their VR’s decrease (increase).

Indeed, the coefficient bd is negative and highly significant at all the considered horizons

τ . The opposite holds for wages: the relation between their VR’s and the labor-share is

positive and the coefficient bw is significant for τ large enough.11 In turn, a positive and

highly significant relation obtains between the labor-share and the distance between the VR’s

of wages and dividends (i.e. bwd > 0). Note that sign and significance of the coefficients are

preserved when controlling for financing and investment decisions. Adjusted R2 are similar

too.12

Figure2The left panel of Figure 2 shows the positive relationship between the labor-share and

the gap between the VR’s of wages and dividends. The middle and the right panels show the

positive and negative relationships between the labor-share and the VR’s of respectively wages

and dividends. The above results strongly support the idea that income insurance within the

firm is a main driver of the timing of dividend risk.

C. Income Insurance and the Term-Structure of Equity

Now the focus turns from macroeconomic fundamentals to the term-structure of equity. As

long as income insurance induces a term-structure effect, I expect that variation in the labor-

share is related with risk and return on short-term assets, such as dividend strips. This could

obtain by means of both the cash-flows and the discount rates channels.

In order to test whether income insurance is a driver of dividend strips returns, I build

quarterly time-series of excess returns and GARCH excess return volatility. Data is monthly

from van Binsbergen et al. (2012) on the sample 1996:4-2010:1 (i.e. the longest available

data).13 To measure the slope of the term-structure of equity, I also build the time-series of

dividend strips returns in excess of market returns. Then, I regress these three time-series on

12

the lagged labor-share and a bunch of macroeconomic and financial controls:

volstript = a + briskW/Yt−1 + b′ccontrolst + εt (short-term equity risk),

rstript − rrisk-freet = a + bpremiumW/Yt−1 + b′ccontrolst + εt (short-term equity premium),

rstript − rmktt = a + bslopeW/Yt−1 + b′ccontrolst + εt (short-term equity slope).

Since income insurance induces a leverage effect on dividends and enhances their short-run

risk, we expect a positive relation between the labor-share and the three dependent variables.

Regression results are reported in Table II. The labor-share significantly and positively

predicts dividend strips volatility one-quarter, one-year ahead and two-years ahead. Newey-

West t-statistics are 5.88, 6.03 and 5.37. Adjusted-R2 are extremely high: 72%, 74% and 62%

(without controls). Note that these variables do not share a time-trend and are not strongly

persistent (the labor-share half-life is about 3.5 years).14

TableIIThe left panel of Figure 3 shows how labor-share shapes dividend strips volatility: both

time-series increase between 1996 and 2001, then they decrease up to 2007, increase again

up to 2010. To understand how explanatory power of labor-share relates with other potential

drivers of equity risk, Panel A of Table III shows horse race regressions. I consider the following

controls: the three Fama and French factors (excess market return, SMB and HML returns),

market return volatility, log price-earnings ratio, risk-free rate, cay as well as financial leverage

and investment-share from the non-financial corporate sector.

Figure3First, labor-share coefficients remains highly significant with all of these controls. Sec-

ond, horse race regressions do not change the positive sign and magnitude of the labor-share

coefficients. Third, adjusted-R2 do not increase substantially in any of the horse race regres-

sions. Thus, the above results suggest that labor-share variation is a very important driver of

short-term equity risk.15

Table II also shows that the labor-share significantly and positively predicts dividend strip

returns in excess of respectively the risk-free rate and the market return. In the former case,

Newey-West t-statistics and adjusted-R2 are respectively 1.99 and 8% (one-quarter ahead),

3.02 and 31% (one-year ahead) and 6.26 and 64% (two-years ahead). In the latter case, Newey-

13

West t-statistics and adjusted-R2 are respectively 1.98 and 10% (one-quarter ahead) and 2.58,

26% (one-year ahead) and 2.43 and 28% (two-years ahead). Time-series are reported in middle

and right panels of Figure 3. Similarly to the case of volatility, Panels B and C of Table III

show horse race regressions with the same set of controls. Again, labor-share coefficients

remain positive and significant in all of the horse race regressions and the adjusted-R2 do not

increase substantially.16

TableIIIOverall, the above analysis strongly suggests that labor-share variation is an important

driver not only of short-term equity risk but also of short-term equity premium and slope.

Income insurance seems to be more informative about short-term equity returns than many fi-

nancial indicators (e.g. valuation ratios and Fama and French factors) and alternative macroe-

conomic channels (e.g. financial and investment decisions).

D. Additional Implications and Robustness

D.1. Income Insurance and Expected Growth

The previous analysis documents how income insurance affects the term structure of both

dividend risk and equity returns. I now look at additional testable implications. As long

as income insurance concerns transitory risk only, the labor-share should positively forecast

dividend growth. This obtains because, on the one hand, the labor-share is a decreasing

function of the transitory component of output and, on the other hand, dividends positively

load on such a transitory shock. Therefore, I perform long-horizon predictability regressions

of dividend growth rates on the labor-share. Estimation results are reported in panel A of

Table IV.

TableIVThe regressions support the mechanism of income insurance. Coefficients are positive

and highly significant (as usual, Newey-West and Hansen-Hodrick t-statistics are used to

account for autocorrelation and heteroscedasticity of residuals and overlapping samples). The

explanatory power is large: the adjusted R2 ranges from about 0% to 37%. The mechanism

of income insurance also implies that the labor-share should positively forecast consumption

14

growth. Panel B of Table IV provides evidence that this is the case. Coefficients are positive

and significant at all the considered horizons. Adjusted R2 ranges from about 6% to 36%.

These results provide further robustness to the mechanism of income insurance, which leads

to dividends and consumption predictability.17

D.2. Other Sources of Transitory Dividend Risk

This section aims to verify whether downward-sloping dividend risk as well as short-term equity

risk, premium and slope could be imputed to sources of transitory dividend risk distinct from

income insurance.

First, I isolate the transitory component of dividends which is not due to income insurance

by looking at the dividend-share orthogonalized with respect to the labor-share. Even if the

negative correlation between the dividend-share and the labor-share is highly significant, the

latter only explains about 8% of variation in the former on the sample 1996:4-2010:1, for which

we have dividend strip returns. Thus, a non-negligible component of transitory dividend risk

should be imputed to sources different from income insurance (such as financial or investment

decisions as well as the dividend policy itself).

Second, I use such an orthogonalized dividend-share as a control in the regressions of the

variance-ratios of dividends at several horizons on the labor-share. If transitory risk distinct

from income insurance drives the timing of dividend risk, we expect a significant coefficient

of the orthogonalized dividend-share as well as an increase in the adjusted R2. Results are

reported in Panel A of Table V. The coefficient of the orthogonalized dividend-share is never

significant and its sign also changes with the horizon. This suggests that transitory risk

distinct from income insurance does not help explaining the timing of dividend risk. Indeed,

adjusted R2 are almost identical to those in Panel C of Table I.

TableVThird, I use the orthogonalized dividend-share as a control in the regressions of dividend

strips return volatility, excess return over the risk-free rate and excess return over the market

return on the current labor-share. Results are reported in Panel B of Table V. The coefficient

of the orthogonalized dividend-share is not significant in 8 out of 9 regressions. Adjusted R2

15

are almost identical to those of the univariate regressions in Table II.

This section shows that, even if transitory dividend risk distinct from income insurance is

non-negligible, it does not contribute to explain either the timing of dividend risk, short-term

equity risk, short-term equity premium or equity slope. This is not surprising by inspection

of the right panel of Figure 1: the decreasing variance ratios of output minus wages almost

recovers the negative slope of dividends. Thus, even if alternative mechanisms to income

insurance could potentially affect the timing of dividend risk, it seems they do not.18 These

results provide further robustness to the term-structure effect of income insurance.

II. Model

A. Economy

A representative firm produces an operational cash-flows, which can be interpreted as the

total output minus investments: C = Y − I. Such an operational cash-flows represents the

total resources shared by workers and shareholders: the former receive wages (W ) and the

latter receive dividends (D). The resource constraint requires C = W +D. To keep the model

simple, I assume limited market participation such that workers do not access the financial

markets and consume their wages. Consequently, shareholders act as a representative agent

on the stock market and consume dividends.19

Agents feature recursive preferences in spirit of Kreps and Porteus (1979), Epstein and

Zin (1989) and Weil (1989). For the sake of tractability, I assume their continuous time

counterpart, as in Duffie and Epstein (1992). These preferences allow for the separation

between the elasticity of intertemporal substitution (EIS) and the coefficient of relative risk

aversion (RRA). Given a consumption process C, the utility at each time t is defined as

Jt = Et∫u≥t

f(Cu, Ju)du, with f(C, J) = βχJ(

C1−1/ψ

((1−γ)J)1/χ− 1), (1)

where χ = 1−γ1−1/ψ

, γ is RRA, ψ is EIS and β is the time-discount rate. Power utility obtains

16

for ψ → 1/γ. Workers’ and shareholders’ consumption levels are denoted by respectively

Cw,t = Wt and Cs,t = Dt.

Total resources are interpreted as a technological shock and their dynamics are modelled

as the product of a permanent and a transitory shock. The former, X, features time-varying

expected growth, in spirit of long-run risk literature, and induces an upward-sloping effect on

the term-structure of risk. The latter, Z, is usually considered in the real business cycle liter-

ature and induces a downward-sloping effect. Then, the two shocks jointly lead to flexibility

at modelling the timing of risk. Namely, total resources C = XZ have dynamics given by:

d logXt = dxt = (µt − σ2x/2)dt+ σxdBx,t, (2)

dµt =λµ(µ− µt)dt+ σµdBµ,t, (3)

d logZt = dzt = − λzztdt+ σzdBz,t, (4)

Those processes feature homoscedasticity and independent Brownian shocks for the sake of

exposition and tractability.

B. Income Insurance, Wages and Dividends

The standard Walrasian Cobb-Douglas economy leads to a constant labor share and workers

paid at their marginal productivity: Wt = αCt, with α ∈ (0, 1). The opposite extreme

scenario would be the case where workers are promised perfect income smoothing, i.e. wages

are constant (and potentially equal to the unconditional expected marginal productivity).

Here, I postulate that workers and shareholders agree in advance on a risk sharing rule

C. The two agent types arrange an agreement such that wages and dividends share the

same long-run dynamics of total resources but the former feature an insurance to transitory

shocks, implying a levered exposition of the latter. Namely, the contingent wage payments

and dividend distributions are governed by the following sharing rule:

C ={

(Wt, Dt) : Wt = Ct ω(zt), Dt = Ct δ(zt), Wt +Dt = Ct ∀t}, (5)

17

where the labor-share and dividend-share satisfy

− 1 < ω′(zt)ω(zt)

< 0, ω′′(zt) > 0 ⇒ δ′(zt)δ(zt)

> 0, δ′′(zt) < 0, (6)

such that both wages and dividends increase with zt but are respectively concave and convex

(i.e. income insurance strengthens in bad times). As it will be shown throughout the paper,

these assumptions capture the mechanism of income insurance about transitory risk as well

as many related empirical patterns documented in the literature and in Section I: (i) Ct, Wt

and Dt are subject to both permanent and transitory shocks, (ii) Ct, Wt and Dt are co-

integrated, (iii) the labor-share is counter-cyclical, (iv) short-term growth rates of wages and

dividends are respectively smooth and levered, (v) VR’s of wages are upward-sloping, (vi)

VR’s of dividends are downward-sloping, (vii) VR’s of total resources are about flat, (viii)

VR’s of wages and dividends move respectively positively and negatively with the labor-share,

(ix) the gap between the VR’s of wages and dividends moves positively with the labor-share,

(x) the labor-share positively forecasts dividends’ growth as well as total resources’ growth,

(xi) the labor-share positively forecasts dividend strip return volatility, premium and slope

under standard assumptions.

For the sake of exposition and tractability, I assume a simple and parsimonious functional

form:

ω(zt) = αe−φzt ⇒ δ(zt) = 1− αe−φzt , (7)

where the curvature parameter φ ∈ (0, 1) represents the degree of income insurance and

α ∈ (0, 1) is the steady-state labor-share (or the constant labor-share in absence of income

insurance for φ→ 0). Therefore, the parameter φ should be interpreted as the joint effect of

bargaining negotiations between workers and shareholders (Danthine and Donaldson (1992,

2002)) as well as labor market frictions (Shimer (2005), Gertler and Trigari (2009), Favilukis

and Lin (2015)) which lead to labor rigidity and operational leverage.

18

Denote with

d log I = µIdt+ σIxdBx + σIz dBz, I = {C,W,D}

the instantaneous dynamics of total consumption, wages and dividends. Provided φ > 0, a

number of results is worth noting. First, long-run growth affects in the same way the drift

(∂µµW = ∂µµ

C = ∂µµD = 1) and volatility (σWx = σCx = σDx = σx) of total consumption,

wages and dividends because labor relations do not alter the impact of permanent shocks.

Second, expected consumption growth depends on zt: |∂zµC | = λz. However, the persistence

of expected growth of wages and dividends is altered by income insurance: the larger φ, the

stronger the persistence of wages (i.e. |∂zµW | < λz) and the weaker that of dividends (i.e.

|∂zµD| > λz). Third, the volatility of wages and dividends due to transitory shocks is affected

by income insurance. Namely, σWz =(1 + ω′(zt)

ω(zt)

)σz = (1−φ)σz is constant, lower than σCz and

decreasing with φ. Therefore, income insurance induces a smoothing effect on wages at the cost

of more volatile dividends. Indeed, σDz =(1 + δ′(zt)

δ(zt)

)σz is larger than σCz and heteroscedastic:

σz = σCz < σDz =(

1 + φ ω(zt)1−ω(zt)

)σz. (8)

Fourth, such an excess volatility or leverage, σDz −σz, is proportional and increasing with both

φ and the labor-share and counter-cyclical (i.e. ∂zσDz < 0).

The focus now turns on the term structure of growth rates’ volatility. Denote with

Ct(τ, u) = Et[Cut+τ ], Wt(τ, u) = Et[W u

t+τ ] and Dt(τ, u) = Et[Dut+τ ] the moment generating

functions of the logarithm of total consumption, wages and dividends. The term structure of

dividends growth rates’ volatility is computed as in Belo et al. (2015):

σ2D(t, τ) =

1

τlog

(Dt(τ, 2)

Dt(τ, 1)2

). (9)

Note that µt and zt lead respectively to an upward-sloping (∂σµ,τσD(t, τ) > 0) and a downward-

sloping effect (∂σz ,τσD(t, τ) < 0). Interestingly, the stronger income insurance, the larger

19

the volatility level (∂φσD(t, τ) > 0) and the more pronounced the downward-sloping effect:

∂φ,τσD(t, τ) < 0. Therefore, income insurance enhances the strength of the transitory shock

and leads to an excess of short-run risk in dividends distributions with respect to total con-

sumption. The opposite holds for wages: ∂φσW (t, τ) < 0 and ∂φ,τσW (t, τ) > 0. This effect is

the model counterpart of the empirical distance between the term-structures of output and

those of dividend risk and wage risk, documented in Figure 1.

While the instantaneous volatility of dividends is increasing with the labor-share as in

Eq. (8), the larger the horizon, the lower the sensitivity of dividend volatility because zt is a

transitory shock. Thus, the variance ratio of dividends,

VRD(t, τ) =σ2D(t, τ)

σ2D(t, 1)

,

is decreasing with the labor-share, as documented in Table I. In turn, the distance between

the variance ratios of wage and dividends –i.e. the term-structure effect of income insurance–

is increasing with the labor-share, as in the actual data.

Similarly, a number of other interesting quantities obtain in closed form. Denote with

σω(t, τ) and σδ(t, τ) the volatilities of the labor-share and dividend-share and with ρC,ω(t, τ)

and ρC,δ(t, τ) their correlations with total consumption. For φ > 0, we obtain:

0 < σω(t, τ) < σδ(t, τ),

ρC,ω(t, τ) < 0 < ρC,δ(t, τ),

such that the labor-share is smoother than the dividend-share, the former is counter-cyclical

and the latter is pro-cyclical. These results are due to income insurance, and are also an

important feature of the data, as shown in Section I.

For the sake of exposition and to derive analytical results for asset prices, in the following

I use a log-linearized version of dividends dynamics:

logDt ≈ logXt + log D + ∂z logD∣∣z=0

zt = xt + d0 + dzzt, (10)

20

where log D = d0 = log(ezt − αe(1−φ)zt

)∣∣z=0

captures the steady-state dividend share 1 − α,

and dz satisfies:

dz = 1 + φ ω(zt)1−ω(zt)

∣∣z=0

= 1 + φ α1−α . (11)

Note that, as usual, the dividend process is still increasing in the states xt and zt but inherits

their homoscedasticity (Section IV considers heteroscedasticity in fundamentals).

C. Asset Prices

C.1. Equilibrium State-Price Density

Preferences in Eq. (1) and the log-linearized dividend dynamics of Eq. (10) guarantee a

model solution which emphasizes the role of the state-variables µt and zt in the formation of

prices. A first order approximation of the shareholders’ consumption-wealth ratio around its

endogenous steady state, ecq, provides closed form solutions for prices and return moments up

to such approximation (Benzoni, Collin-Dufresne, and Goldstein (2011)).

From the shareholders’ perspective, the marginal utility evaluated at optimal consumption

is the valid state-price density (Duffie and Epstein (1992)):

ξt,u = e∫ ut fJ (Cs,τ ,Js,τ )dτ fC(Cs,u,Js,u)

fC(Cs,t,Js,t), ∀u ≥ t. (12)

Hence, the price of an arbitrary payoff stream {Fu}u≥t is given by Et[∫∞tξt,uFudu]. The

equilibrium state price density ξ0,t satisfies:

Ct = Wt +Dt = IC [ξ0,t] e−d0−(dz−1)zt , (13)

where IC [ξ0,t] ={Cs,t : ξ0,t = e

∫ t0 fJ (s)dsfC(t)

∣∣Xt, µt, zt}

denotes the time t shareholders’

optimal consumption implied by ξ0,t.

Although the state price density equals the first order condition of shareholders, it depends

on the risk attitudes of both agent types. The left hand side of Eq. (13) is given by the

21

exogenous flows of total resources produced by the firm. The right hand side is given by the

product of two terms: the former is the optimality condition for market participants; the latter

is a time-varying term which captures the distributional risk due to income insurance. Namely,

this term equals the inverse of the dividend-share, Ct, /Dt, and its variability increases with

φ. Instead, in absence of income insurance, the labor-share is constant and the second term

on the right hand side of Eq. (13) reduces to (1− α)−1.20

Note that, even if the equilibrium state-price density is an involved function of the in-

tegrated process Ct, the economy is characterized by a stationary equilibrium. This is a

necessary condition to produce realistic testable implications. Many real business cycle mod-

els circumvent the problem by excluding permanent shocks (e.g. Ct = Zt). Here, allowing

for both permanent and transitory shocks (e.g. Ct = XtZt) and still obtaining a stationary

equilibrium is of crucial importance in order to study the equilibrium implications for the

term-structure of equity.

The equilibrium state price density has dynamics given by

dξ0,tξ0,t

= dfCfC

+ fJdt = −r(t)dt− θx(t)dBx,t − θµ(t)dBµ,t − θz(t)dBz,t, (14)

where the instantaneous risk-free rate satisfies

r(t) = r0 + rµµt + rzzt, (15)

with rµ = 1ψ

, rz = −λzψdz and the instantaneous prices of risk are given by

θx(t) = − ∂XfCfC

Xtσx = γσx, (16)

θµ(t) = − ∂µfCfC

σµ = γ−1/ψecq+λµ

σµ, (17)

θz(t) = − ∂zfCfC

σz =(γ − λz(γ−1/ψ)

ecq+λz

)dzσz. (18)

The risk-free rate is affine in µt and zt. The coefficients rµ and rz are respectively positive

and negative and both decrease in magnitude with ψ, as usual under recursive preferences.

22

Moreover, rz increases in magnitude with the degree of income insurance φ.

The permanent shock commands a price of risk θx(t) due to the contribution of xt to the

instantaneous volatility of shareholders’ consumption. Long-run growth commands a price of

risk θµ(t), which has the usual form in long-run risk models. This is a price for the contribution

of µt to the variation in the continuation utility value of shareholders. Hence, θµ(t) disappears

under power utility (ψ → γ−1). Such a price of risk increases with the preferences for the early

resolution of uncertainty γ−1/ψ and decreases in magnitude with the rate of reversion λµ. The

transitory shock leads to a price of risk θz(t), which has two components. The first, γdzσz,

is a positive price for the contribution of zt to the instantaneous volatility of shareholders’

consumption. The second, θz(t)− γdzσz, is a price for the contribution of zt to the variation

in the continuation utility value of shareholders. Namely, at the opposite of θµ, this term is

negative and increasing in the rate of reversion λz under preferences for the early resolution

of uncertainty. This term disappears under power utility. Interestingly, both the components

of θz(t) are proportional to the coefficient dz > 1, which is increasing in φ. Therefore, for

γ > ψ > 1, income insurance leads to a positive price for its effect on the current dividend

volatility and a negative price for its effect on the evolution of shareholders’ utility.

C.2. Equilibrium Dividend Strips

The equilibrium price of the market dividend strip with maturity τ is exponential affine in

xt, µt and zt:

Pt,τ = Et [ξt,t+τDt+τ ] = XteA0(τ)+Aµ(τ)µt+Az(τ)zt , (19)

Hence, the strip’s price-dividend ratio is stationary. The functions Aµ(τ) and Az(τ) are

respectively the semi-elasticity of the price with respect to µt and zt:

Aµ(τ) = ∂µ logPt,τ = (1−e−λµτ )(1−1/ψ)λµ

, (20)

Az(τ) = ∂z logPt,τ =(

(1− e−λzτ ) + e−λzτ)dz. (21)

23

First, Aµ(τ) increases with ψ, whereas Az(τ) decreases with ψ. Second, Az(τ) increases with

the degree of income insurance φ. Therefore, the leverage effect on dividends due to income

insurance also affects prices: this effect is amplified for ψ < 1 and vice-versa. Finally, for

ψ → 1, the strip’s price-dividend ratio reduces to a state-independent function of the horizon

τ . The instantaneous volatility and premium on the dividend strip with maturity τ are given

by

σP (t, τ) =

√σ2x + (1−e−λµτ )2(ψ−1)2

λ2µψ2 σ2

µ + e−2λzτ (ψ+eλzτ−1)2

ψ2 d2zσ

2z , (22)

(µP − r)(t, τ) = γσ2x + (1−e−λµτ )(1−1/ψ)(γ−1/ψ)

λµ(ecq+λµ)σ2µ +

(1ψ

(1−e−λzτ )+e−λzτ)

(λz+ψγecq)

ψ(ecq+λz)d2zσ

2z . (23)

The three shocks of the model contribute to the return volatility and command a premium.

The permanent shock does not lead to excess volatility. Instead, the loadings on Bµ,t and Bz,t,

are proportional to σµ and σz, but also depend on the horizon τ , the elasticity of intertemporal

substitution and the persistence of the states. Namely, the loading on long-run growth is

increasing with ψ and decreasing with λµ. Instead the loading on the transitory shock is

decreasing with ψ and increasing with λz. The latter is also amplified by the leverage effect

due to income insurance, dz.

The premium on the dividend strip is given by the sum of the compensations to the three

shocks. The compensation for the permanent shock is positive and has the usual form: γσ2x.

Instead, the compensations associated to the states µt and zt depend also on the horizon τ ,

the elasticity of intertemporal substitution and the persistence of the states. Long-run growth

commands a premium which is increasing with ψ and decreasing with λµ:

γ−1/ψλµ(ecq+λµ)

Aµ(τ)σ2µ. (24)

This term is a compensation for the contribution of µt to the variability of the continuation

utility value of shareholders. If the intertemporal substitution effect dominates the wealth

effect and shareholders have preferences for the early resolution of uncertainty –e.g. γ > ψ >

24

1– long-run growth leads to a positive premium on the dividend strip. The premium for the

exposition to the transitory shock zt is given by the sum of two terms:

γAz(τ)dzσ2z , and − λz(γ−1/ψ)

ecq+λzAz(τ)dzσ

2z . (25)

The first term compensates for the contribution of zt to the instantaneous volatility of share-

holders’ consumption. Such a premium is always positive and decreases with ψ. Instead, the

second term is an intertemporal compensation. It is decreasing with ψ and is negative (posi-

tive) under the shareholders’ preference for the early (late) resolution of uncertainty. Moreover,

both terms depend on the horizon τ and increase in magnitude with income insurance, dz.

The following is a key result of the paper. The slopes of the term-structures of dividend

strips’ volatility and premia are given by

∂τσ2P (t, τ) = 2(ψ−1)

ψ2

(e−2λµτ (eλµτ − 1)(ψ − 1)λ−1

µ σ2µ − e−2λzτ (ψ + eλzτ − 1)λzd

2zσ

2z

), (26)

∂τ (µP − r)(t, τ) = ψ−1ψ2

( (γ−1/ψ)e−λµτσ2µ

ecq+λµ− (λz+γψecq)e−λzτλzd2zσ

2z

ecq+λz

). (27)

The slope of the term-structure of volatility depends on two terms, due respectively to µt and

zt. The first is always positive and, hence, implies an upward sloping effect. The second term

is negative if the intertemporal substitution effect dominates the wealth effect and vice-versa.

Therefore, the term-structure of volatility is monotone upward sloping if ψ < 1, whereas it is

not necessarily monotone if ψ > 1. A non-monotone (e.g. U-shaped) term-structure of risk

obtains if the leverage effect due to income insurance, dz, outweighs the upward sloping effect

due to long-run growth for some horizons τ .

Also the slope of the term-structure of premia depends on two terms, due to µt and zt. The

first is positive if the intertemporal substitution effect dominates the wealth effect (ψ > 1) and

shareholders have preferences for the early resolution of uncertainty (γ > 1/ψ). The second

term is negative if the intertemporal substitution effect dominates the wealth effect (ψ > 1)

and vice-versa. Thus, the slope of equity premia can be economically interpreted as:

25

slope =intertemporal

substitution︸ ︷︷ ︸ψ−1

×(

resolution of

uncertainty︸ ︷︷ ︸γ−1/ψ

× long-run

risk︸ ︷︷ ︸λµ, σµ

− income

insurance︸ ︷︷ ︸φ

× short-run

risk︸ ︷︷ ︸λz , σz

).

Under the usual parametrization γ > ψ > 1, variation in long-run growth leads to an upward-

sloping effect, whereas transitory shocks lead to a downward-sloping effect. Therefore, the

term-structure of equity premia is not necessarily monotone as long as both permanent and

transitory shocks enter the model. Namely, the slope is negative for income insurance strong

enough.21

Such an analytical result clearly explains why the standard long-run risk model cannot

capture the recent evidence about dividend strips. Long-run risk models (i.e. Ct = Xt) rule

out transitory shock zt and do a good job at matching a number of asset pricing moments as

long as ψ > 1. Both the term-structures of volatility and premia are monotone increasing.

The alternative scenarios, in which either only the transitory shock enters the model (Ct = Zt)

or long-run growth is constant (Ct = XtZt with µt = µ), lead to monotone decreasing term-

structures of risk and premia for ψ > 1. However, in such cases the equity premium is not

sizeable since it decreases with ψ. Instead, when we account for both µt and zt, the model

can accommodate for both a high equity premium and downward sloping term-structures of

equity risk and premia in the short-run. Therefore, the dynamics of dividends induced by

income insurance reconciles the above stylized facts.

C.3. Equilibrium Market Asset

The market price is given by the time integral of the dividend strip price, Pt =∫∞

0Pt,τdτ :

Pt = Et[∫∞tξt,uDudu

]= Xtβ

−1eu0χ−1+d0+uµχ−1µt+

(uzχ−1+dz

)zt (28)

where u0, uµ and uz are endogenous constants. The market price dividend ratio is a stationary

function of µt and zt and equals the wealth-consumption ratio of shareholders. Prices increase

with µt as long as the intertemporal substitution effect dominates the wealth effect and increase

with zt: ∂µPt ≷ 0 if ψ ≷ 1, ∀γ and ∂zPt > 0, ∀ψ, γ.

26

The instantaneous volatility and premium on the market asset are given by

σP (t) =

√σ2x +

(1−1/ψecq+λµ

)2

σ2µ +

(1− (1−1/ψ)λz

ecq+λz

)2

d2zσ

2z , (29)

(µP − r)(t) = γσ2x + (1−1/ψ)(γ−1/ψ)

(ecq+λµ)2σ2µ + (γψecq+λz)(ψecq+λz)

(ecq+λz)2ψ2 d2zσ

2z . (30)

The permanent shock xt enters the return dynamics exactly as the dividend dynamics, σx.

Instead, the loadings on the shocks Bµ,t and Bz,t depend on the preference parameters and are

respectively increasing and decreasing in ψ. The loading on Bµ,t always leads to an excess-

volatility of market returns over dividends, whereas the loading on Bz,t generates excess-

volatility for ψ < 1 and vice-versa.

The equity premium is given by three components due to the three shocks. The per-

manent shock xt requires the usual positive compensation γσ2x. Long-run growth leads to a

premium, which is positive if the intertemporal substitution effect dominates the wealth effect

and shareholders have preference for the early resolution of uncertainty. Instead, zt commands

a premium which is always positive, decreasing with ψ and increasing with γ. Furthermore,

this compensation term is increasing in the degree of income insurance.

Finally, provided γ > ψ > 1, the whole equity premium is increasing in γ but non-monotone

in ψ. Indeed, µt leads to compensations increasing with the horizon, whereas short-run but

persistent uncertainty due zt leads to compensations decreasing with the horizon. Once the

model is calibrated with realistic parameters, such a term-structure perspective sheds lights

on which risks are priced in equilibrium. Using the definition of the market asset price as

the time integral of the dividend strip prices, it is possible to write the equity premium as a

time integral and, hence, to derive an equilibrium density which describes the contribution of

future discounted cash-flows at each horizon (details are in the Appendix):

H(t, τ) =Π(t, τ)

(µP − r)(t)with (µP − r)(t) =

∫ ∞0

Π(t, τ)dτ =

∫ ∞0

Pt,τPt

(µP − r)(t, τ)dτ.

(31)

The shape of the equilibrium density H(t, τ) tells us whence the equity premium comes from.

27

The more the mass of probability concentrates on either short or long horizons, the more the

riskiness associated to such horizons deserves a compensation and contributes to the whole

equity premium. Therefore, H(t, τ) is natural metric to understand the effect of the term-

structures of equity on an important equilibrium outcome, such as the equity premium.22

C.4. Equilibrium Bond and Equity Yields

The equilibrium price of the zero-coupon bond with maturity τ , Bt,τ = Et [ξt,t+τ ], is stationary

and exponential affine in µt and zt . Hence, the bond yield is state-dependent but its volatility

inherits the homoscedasticity of the states:

ε(t, τ) = − 1τ

logBt,τ = 1τ

(−K0(τ) + (1− e−λµτ )rµλ−1

µ µt + (1− e−λzτ )rzλ−1z zt

), (32)

where K0(τ) is a deterministic function of the maturity. The short- and long-run limits of the

term-structure of real yields lead to the steady state term-spread:

ε(t,∞)− ε(t, 0) = − rµθµσµλµ− r2µσ

2λ2µ− rzθzσz

λz− r2zσ

2z

2λ2z, (33)

which can be either positive or negative for γ > ψ > 1: indeed, rµθµ > 0 and rzθz < 0.23

Armed with these results, the focus turns on the equity yields as introduced by van Bins-

bergen et al. (2013): p(t, τ) = 1τ

logDt/Pt,τ . The model equity yield is given by

p(t, τ) = 1τ

(d0 − A0(τ) + (1−e−λµτ )(1/ψ−1)

λµµt + (1− e−λzτ )(1− 1/ψ)dzzt

)(34)

and, hence, is a stationary function of the states and the maturity. Moreover, it can be

decomposed as:

p(t, τ) = ε(t, τ)− gD(t, τ) + %(t, τ). (35)

The equity yield is given by the difference among the yield on the risk-less bond, ε(t, τ), and

the dividend expected growth, gD(t, τ) = log(Dt(τ, 1)/Dt)/τ , plus a premium, %(t, τ). The

28

latter is a state-independent function of the maturity. The transitory shock determines the

level of its short-run limit, whereas long-run growth determines the level of its long-run limit.

Therefore, the term-spread

%(t,∞)− %(t, 0) = γψλµ+ecq

(λµ+ecq)λ2µψσ2µ −

λz+γψecq

(λz+ecq)ψd2zσ

2z ,

can be either positive or negative depending on the model parameters. Thus, downward-

sloping premia on equity yields obtain for income insurance large enough.

III. Asset Pricing Results

A. Model Calibration and the Timing of Dividend Risk

Model parameters are set by choosing cash-flows parameters in order to match key moments

from the time-series of consumption, wages and dividends growth rates and by choosing pref-

erence parameters to provide a good fit of standard asset pricing moments.

The calibration procedure uses information from the term-structures of cash-flows to assess

the equilibrium asset pricing implications. Namely, I exploit analytical solutions to set the

cash-flows parameters. The model has eight parameters Θ = {µ, σx, λµ, σµ, λz, σz, α, φ}, which

characterize the joint dynamics of aggregate consumption, wages and dividends. I choose eight

moment conditions: the relative error between the empirical and the model long-run growth

of consumption (gC), yearly volatility of consumption (σC(1)) and dividends (σD(1)) growth

29

rates, average level of the dividend-share (δ) and its volatility (σδ):

m1(θ) =|gempiricalC −gC |gempiricalC

,

m2(θ) = |σC(1)empirical−σC(1)|σC(1)empirical ,

m3(θ) = |σD(1)empirical−σD(1)|σD(1)empirical ,

m4(θ) = |δempirical−δ|δempirical ,

m5(θ) =|σempiricalδ −σδ|σempiricalδ

,

and three additional conditions that capture the relative error between the empirical and the

model term-structures of variance ratios of consumption, wages and dividends over a 15 years

horizon:

m6(θ) =∑15

τ=2

|V RempiricalC (τ)−V RC(τ)|V Rempirical

C (τ),

m7(θ) =∑15

τ=2

|V RempiricalW (τ)−V RW (τ)|V Rempirical

W (τ),

m8(θ) =∑15

τ=2

|V RempiricalD (τ)−V RD(τ)|V Rempirical

D (τ).

The variance-ratios of growth rates are computed as V Ri(τ) =σ2i (τ)

σ2i (1)

for i = {C,W,D}.

The latter three moment conditions capture the timing of the macroeconomic risk and, in

particular, the term-structure effect of income insurance.

Finally, I obtain the parameter vector Θ by minimizing the average-relative-error:

Θ = arg minθ

ARE(Θ) = arg minθ

18

∑8

i=1mi(θ).

The empirical moments are as follows: I set the long-run growth rate of consumption to 2%

and the volatility of consumption to 2.5%, which are the usual values from the literature;

the volatility of dividends is set to 17.4%, the average value of the dividend-share (i.e. the

ratio of net dividends over the sum of net dividends and wages) is set to 6% and its volatility

to 1.6%. These values are from the US non-financial corporate sector and are close to the

30

values considered in Longstaff and Piazzesi (2004), Lettau and Ludvigson (2005) and Santos

and Veronesi (2006). The variance-ratios of wages and dividends from the US non-financial

corporate sector are computed as in Section I: wage risk increases from one to about 1.6

over a 15 years horizon, whereas dividend risk decreases from one to about 0.2. Finally, the

variance-ratios of consumption (NIPA nondurable goods and services) are about flat. Table VI

reports the model parameters and Table VII reports both the empirical and the model-implied

moments of cash-flows.

TableVIThe left panel of Figure 4 shows the model implied term-structures of variance-ratios for

both aggregate consumption, wages and dividends, as well as their empirical counterparts.

The model accurately captures both rise and decline of respectively wage and dividend risk

with the horizon. Therefore, the calibration procedure does a good job at recovering the

whole shape of the empirical term-structures and, hence, the timing of consumption, wages

and dividend risk.

TableVIIThe right panel of Figure 4 shows that empirical and model-implied autocorrelation func-

tions of the labor-share are very similar. This is interesting because, even if autocorrelations

do not enter the calibration procedure, the capability of the model to match the gap in the

variance ratios of wages and dividends leads to an accurate description of the persistence of

the main model state-variable (half-life is about 3.5 years).

Figure4The decline in the timing of dividend risk is due to the levered exposition of dividends to the

transitory component of consumption, zt. The operating leverage coefficient, dz = 1 +φ α1−α =

6.5, allows for both i) the correct slope of the term-structure of dividends volatility, and ii)

the excess volatility of dividends over consumption in yearly growth rates (i.e. 17.4% versus

2.5%). At the same time, the model leads to smooth wage growth rates (i.e. 1.7%).

Figure5Section I empirically investigates the term-structure effect of income insurance by recog-

nizing a dynamic relationship between the labor-share and, respectively, the variance ratios

of dividends and the differential in the variance ratios of wages and dividends. The former

relationship is negative and the latter is positive, as documented in Table I. The model leads

to the same result: the left and right panels of Figure 5 show that the economic mecha-

31

nism of income insurance is the driver of heterogeneity in the timing of macroeconomic risk.

When wages are high (low) relative to dividends, as a result of income insurance, dividend

risk becomes more (less) downward-sloping and the gap between the timing of wage risk and

dividend risk increases (decreases). Similarly, income insurance leads to dividend and con-

sumption growth predictability. Table IV documents a positive intertemporal relationship

between the current labor-share and future growth. The same result holds in the model, as

shown in Table VIII. Since income insurance only concerns transitory risk, both the labor-

share and expected growth are decreasing in the current level of the transitory shock. Hence,

dividend and consumption growth predictability obtain.

TableVIIIA number of insights are noteworthy. First, although very parsimonious, the model dy-

namics are flexible enough to capture the main properties of the empirical data. Thus, the

calibration supports the model assumption of both permanent and transitory shocks. Sec-

ond, income insurance leads to heterogeneity in the term-structures of risk of consumption,

wages, and dividends. Consistently with the empirical findings of Section I, matching the pos-

itive slope of wage risk allows to quantitatively fill the gap between the approximatively flat

term-structure of consumption risk and the downward-sloping one of dividend risk. Thus, the

calibration further supports the idea that income insurance is the main determinant of the tim-

ing of macroeconomic risk. In absence of income insurance (φ = 0), the three term-structures

are equal and collapse on that of consumption. Third, the shape of the term-structure of

dividend risk is the result of the combination of a downward-sloping effect due to zt and an

upward-sloping effect due to µt. One issue with asset pricing models is that often the main

model mechanism essentially relies on a latent factor which is difficult to estimate. A cal-

ibration procedure which exploits the information implied by the term-structures of risk of

macroeconomic variables i) allows to capture additional empirical moments, and ii) offers a

way to infer about the persistence (λµ, λz) and the variance (σµ, σz) of latent factors. Fourth,

the long-run growth factor µt has moderately persistent and smooth dynamics (λµ = .86 and

σµ = 1.8%). The model does not require an excessive persistence in expected consumption

growth, as suggested by Constantinides and Ghosh (2011) and Beeler and Campbell (2012),

32

but accurately captures the persistence of the labor-share.

Finally, I set shareholders’ preference parameters such that they have preference for the

early resolution of uncertainty (γ > 1/ψ) and the intertemporal substitution effect dominates

the wealth effect (ψ > 1), as in most of the asset pricing literature. Namely, in the baseline

calibration the pair γ = 10 and ψ = 1.5 belongs to the usual range of values. The time-discount

rate β is set to 3.5%.

B. The Term Structure of Equity

In the baseline calibration (γ = 10, ψ = 1.5), the term structures of both premia and return

volatility on the dividend strips are decreasing with the maturity at short and medium horizons

–in which the downward-sloping effect due to the transitory shock dominates the upward-

sloping effect due to long-run growth– and approximatively flat at long horizons –in which the

two effects offset each other. The upper panels of Figure 6 show these term structures.

Figure6The stronger the income insurance (φ), the larger the operating leverage effect on dividends

(dz). This enhances the price associated to transitory risk and, for ψ > 1, induces a downward-

sloping effect on the term-structure of equity premia. These are decreasing over a longer

horizon and their slopes are larger in magnitude.

Although the model is parsimonious and data about dividend strip returns are available

on a short sample, it is instructive to compare the slope of risk premia in the model and in

the data. van Binsbergen et al. (2012) verify that one cannot reject the hypothesis that the

one-year dividend strip premium over the market is equal to zero. However, van Binsbergen

and Koijen (2016) propose a more powerful test and verify that the average dividend strip

premium up to 5 years maturity is significantly larger than the equity premium. Thus, the

test provides empirical evidence of a negative slope on the sample 2002-2014. The magnitude

of this premium in U.S. is relatively small, about 1.2% (i.e. 0.1% monthly). The analogous

premium using dividend strip futures returns over the market return in excess of a ten year

bond return is somewhat larger, about 4.1% (i.e. 0.34% monthly). I construct similar ‘spot’

33

and ‘futures’ measures of equity slope in the model:

(spot) 15

∫ 5

0(µP − r)(τ)dτ − (µP − r) = 1.1%,

(futures) 15

∫ 5

0[(µP − r)(τ)− (µB − r)(τ)]dτ − [(µP − r)− (µB − r)(10)] = 3.4%,

where (µP − r) is the equity premium, (µP − r)(τ) and (µB − r)(τ) are the premia on the

dividend strip and the risk-less bond with maturity τ . The baseline calibration produces

values (1.1%, 3.4%) that are very close to their empirical counterparts. The corresponding

values for ψ = 1.25 and ψ = 1.75 are respectively (0.7%, 3.7%) and (1.4%, 3.1%): a higher

elasticity of intertemporal substitution leads to lower equity (higher bond) compensation but

steeper equity (flatter bond) slope.

The lower left panel of Figure 6 shows the term-structure of Sharpe ratios. van Binsbergen

and Koijen (2016) provide evidence that Sharpe ratios from dividend strip futures returns are

slightly increasing with the maturity but are much larger than the Sharpe ratio of market

returns in excess of the 10 year bond yield. Namely, futures Sharpe ratios from 1 to 7 years

maturity range from 42% to 58% in actual data (i.e. from 12% to 17% monthly), whereas

the Sharpe ratio of the market return in excess of the 10 year bond is about 12.5% (i.e. 3.7%

monthly). The corresponding model quantities have similar shapes: futures Sharpe ratios

from 1 to 7 years maturity range from 35% to 38.5%, whereas the Sharpe ratio of the market

return in excess of a 10 year bond is about 7.7%. Thus, both empirical and model short-term

Sharpe ratios are about five times larger than the market Sharpe ratio.24

Dividend strip returns feature excess volatility over dividends at any horizon. Volatility

approaches a value above fundamentals’ risk in the long-run, as shown in the lower right panel

of Figure 6. In particular, the “long-run” excess volatility is given by:

σ2P (t,∞)− σ2

D(t,∞) =(1−2ψ)σ2

µ

λ2µψ2 + d2zσ

2z

ψ2 .

Long-run excess-volatility increases with φ, decreases with ψ and does not depend on risk

aversion. Long-run growth contributes negatively to the excess-volatility for ψ > 1/2 and

34

vice-versa. Instead, zt always contributes positively to the excess-volatility. Thus, for φ large

enough, a positive excess-volatility can obtain even if ψ > 1/2.

The model generates a long-run excess volatility of equity over dividends in line with the

recent empirical evidence about the decreasing variance ratios of dividends, documented by

Belo et al. (2015) and in Section I. Instead, long-run risk models imply payouts to shareholders

riskier than equity returns, as pointed out by Beeler and Campbell (2012). Namely, a long-run

risk model (Ct = Xt) produces “long-run” excess volatility only for ψ < 1/2 –a parametrization

under which standard asset pricing implications fail to obtain.

C. The Market Asset and the Horizon Decomposition of the Equity Premium

The model reconciles the evidence about the term-structures of both equity and macroeco-

nomic variables with standard asset pricing facts. The baseline calibration (γ = 10, ψ = 1.5)

produces an unconditional equity premium, which is about 5.1%, quite close to the data.

Such a result is particularly remarkable since neither stochastic volatility and disasters nor

unrealistically high and time-varying risk aversion are required. The return volatility is about

13.6%, which is somewhat lower than in actual data. However, a slightly lower ψ = 1.25 in-

creases return volatility to 15.8%. Moreover, return volatility leads to a large excess-volatility

over consumption and dividends on the whole term-structure, including the long-horizon. The

model produces a Sharpe ratio (about 37.2%), which accurately matches the data. Such a

result is peculiar and mostly due to transitory risk, amplified by income insurance. The model

leads to a risk-free rate level (about 0.9%) and volatility (about 2.8%), very close to actual

data. The 20 years real bond yield is 3.6%, about 200 basis points higher than TIPs data. The

model also captures quite well the level and the volatility of the price-dividend ratio (about

exp(3.5) and 41.3%).

TableIXTable IX summarizes the empirical and the model-implied moments and reports the asset

pricing moments for many pairs (γ, ψ). A low risk-free rate and high levels of the first two

return moments of the market asset also obtain for (γ = 10, ψ = 1.25), (γ = 7.5, ψ = 1) and

(γ = 5, ψ = 0.75). Decreasing the elasticity of intertemporal substitution leads to an increase

35

in the equity premium but the price-dividend ratio and the risk-free rate are respectively too

smooth and too volatile in comparison with the data. Hence, the choice (γ = 10, ψ = 1.5)

seems preferable. The operating leverage effect due to income insurance not only is crucial

to the modeling of the term-structures but also helps to match the standard asset pricing

moments (Danthine and Donaldson, 2002). The premium increases with φ and decreases with

ψ, all else being equal. The latter relation has the following rationale. The correct calibration

of the timing of macroeconomic risk implies that in equilibrium the price for transitory risk

dominates the price of long-run variations in expected growth.

The case of no income insurance (φ = 0) and, hence, constant labor-share fails to describe

both the risk-free rate and equity premium puzzle as well as the downward-sloping term-

structure of equity.

To better understand the equilibrium equity premium, it is instructive to adopt a term-

structure perspective. The equity premium is a weighted average of dividend strip premia,

where the weights are given by the ratio of the dividend strip price over the equity price. From

the upper left panel of Fig. 6 we observe that dividend strip premia range from about 6.9%

to 4.8%. Since the slope is negative but relatively tiny, a high equity premium (5.1%) can

obtain.25 The equilibrium density H(t, τ) of Eq. (31) represents the relative contribution of

each horizon τ to the whole premium. The horizon decomposition of the equity premium does

not depend only on the dynamics of the permanent and transitory shocks, but also on the

equilibrium compensations for these risks. Figure 7 shows that, in the baseline calibration,

the equilibrium density is monotone decreasing with the horizon. A stronger (weaker) income

insurance mechanism leads to a shift of the density towards the short (long) horizon. Indeed,

income insurance leads not only to riskier dividends but also to a larger compensation for

the transitory risk. This horizon decomposition sheds lights on the nature of priced risk

in equilibrium. A “short-run” explanation of market compensations is needed in order to

simultaneously describe both the standard asset pricing moments and the term-structures of

both fundamentals and equity.

Figure7

36

D. Bond and Equity Yields

Income insurance affects the equilibrium equity yields and their components: the yields on

the risk-less bond, the expected dividend growth and the premium on the equity yields, as

defined by van Binsbergen et al. (2013). Figure 8 plots these quantities at the steady-state as

a function of the horizon both in presence and absence of income insurance.

Figure8The left upper panel shows that the equity yield is decreasing with the horizon under

income insurance, whereas it is increasing in absence of income insurance (φ = 0). For ψ > 1,

the downward-sloping effect due to income insurance outweighs the upward-sloping effect due

to long-run growth. The right upper panel shows the term-structure of the real bond yields.

In absence of income insurance, µt drives the term-structure of interest rates. Similarly to

long-run risk models, a downward-sloping term-structure obtains for ψ > 1, inconsistently

with actual data from TIPs. Instead, in presence of income insurance, zt induces a stronger

upward-sloping effect, which leads to a positive slope for the bond yields. Intuitively, income

insurance enhances transitory risk in equilibrium and, hence, bonds with short maturities are

a better hedge than equity. Therefore, increasing bond yields obtain. Real yields (TIPs) are

available for maturities of 5, 7, 10 and 20 years on the sample 2003-2015.

real bond yields term spreads

% 5y 7y 10y 20y ∞ 7y - 5y 10y - 5y 20y - 5y

TIPs data 0.66 0.96 1.22 1.53 − 0.32 0.57 0.88

Model 2.20 2.54 2.92 3.65 4.37 0.34 0.71 1.35

The baseline calibration produces quite a realistic slope but overestimates the level of the

curve (slightly lower β and/or higher ψ improve the fit at the cost of a lower equity premium).

As commented in the Introduction, the assumption of limited market participation leads to a

state-price density based on dividends. While this seems to be key to capture the short-run

properties of equity, it also creates a tension concerning the quantitative predictions about

real yields. Finally, the right lower panel of Figure 8 reports the term-structure of the equity

yields premia. Such premia increase and decrease with horizon respectively in absence and in

presence of income insurance. For maturities large enough, the equity yield is smaller than the

real yield or, equivalently, the premium on the equity yield is smaller than expected dividend

37

growth. This case leads to negative forward equity yields:

f(t, τ) = p(t, τ)− ε(t, τ) = %(t, τ)− gD(t, τ).

Note that negative forward equity yields are not at odds with actual data, as documented by

van Binsbergen et al. (2013), and do not imply a low equity premium. As long as the negative

slope of %(t, τ) is steep enough, short-term risk is sufficient to produce a high compensation

for equity. This is the case of the baseline calibration (recall that the equity premium is about

5%) and is consistent with the findings of van Binsbergen et al. (2012).

van Binsbergen and Koijen (2016) provide empirical evidence that forward equity yield

volatility decreases with the horizon. Namely, in U.S. 2002-2014 data this volatility ranges

from about 11% to 4% over 1 to 7 years maturities. The corresponding volatility in the baseline

model calibration is also decreasing and ranges from about 15.7% to 5.2%.

In summary, the decomposition of equity yields provides two noteworthy results. First,

income insurance helps to understand the term-structures of the equity yield premia, in line

with the findings of van Binsbergen et al. (2013). Second, at the same time, income insurance

helps to reconcile the term-structure of real interest rates with the equity results.

IV. Uncertainty and Time-Varying Slope of Risk

Premia

While downward-sloping dividend risk is a very robust feature of the data, the empirical

evidence about the term structures of equity is under debate.26 On the one hand, equity

volatility inherits the negative slope of dividend risk; on the other hand, the slope of equity

premia depends on how and how much the sources of long-run and short-run risks are priced

in equilibrium. Downward-sloping equity premia documented by van Binsbergen et al. (2012)

appear as an unconditional property of dividend strips’ returns. Indeed, van Binsbergen et

al. (2013) and Aıt-Sahalia et al. (2015) provide evidence that the term-structure of equity

38

premia is time-varying and switches from flat or slightly increasing in good times to strongly

decreasing in bad times.

The model of the previous sections does not allow to study risk premia time-variation

neither for dividend strips nor for the market asset. While income insurance generates en-

dogenous heteroscedasticity (see Eq. (8)), the log-linearized dynamics of dividends used to

solve for the equilibrium rules out time-variation in equity premia. This section shows that a

minimal modification of the previous model accounts for the refinements by van Binsbergen

et al. (2013). Namely, I introduce heteroscedasticity in the transitory shock zt:

zt = z − zt with dzt = λz(z − zt)dt+ σz√ztdBz,t. (36)

where σz = σz/√z and z > 0. Thus, zt is still a zero-mean stationary process but its volatility

is decreasing in zt. This property captures larger uncertainty during business cycle downturns.

Given the affine specification of the transitory shock, the model can be solved with the same

methodology of the previous sections (proofs are in the Online Appendix B).

This form of heteroscedasticity in fundamentals leads to two results. First, under income

insurance the wage-share is decreasing in zt and, hence, dividends load more on the transitory

shock when the latter is high volatile –i.e. operating leverage is counter-cyclical. Second, the

price of risk associated to transitory risk is no more constant but is decreasing with zt and,

hence, higher compensations obtain during business cycle downturns. These two facts lead to

a slope of the term-structure of equity premia which is time-varying. Long-run growth leads

to an upward-sloping effect and the transitory shock leads to a downward-sloping effect, but

the latter depends on the volatility of zt. Therefore, the downward-sloping effect is counter-

cyclical. As a result, the term-structure of equity premia can be flat or slightly upward-sloping

in good times and strongly downward-sloping in bad times.

To provide an illustration of the equilibrium results, I calibrate the model with the same

procedure of the previous section. For β = 4.5%, γ = 7.5, ψ = 1.5, the model generates steady

state moments in line with the data: 1.1% risk-free rate with 1.7% volatility, an equity premium

39

of 6.0% with return volatility of 14.5% and Sharpe ratio of 41.5% and log price-dividend ratio

of 3.20 with 41.5% volatility.

Figure9The left upper panel of Figure 9 shows the term-structures of the model implied VR’s

(solid lines) and their empirical counterparts (dashed lines). Similarly to Figure 4, the model

matches the upward- and downward-sloping shapes of wage and dividend risk and the flat

term-structure of consumption risk. The right upper panel of Figure 9 shows the model

implied volatility of dividends at the steady-state (solid line) and the 90% probability interval

(dot-dashed and dashed lines). Dividend volatility is strongly decreasing with zt at short

horizons, whereas the effect of the transitory shock disappears in the long-run. The lower

panels of Figure 9 shows the model volatility (left) and premia (right) of the dividend strip

returns at the steady-state (solid line) and the 90% probability interval (dot-dashed and dashed

lines). Under standard preferences (γ > ψ > 1), equity volatility inherits the negative slope of

dividend risk. The transitory shock affects the level of the term-structure, which is counter-

cyclical, but does not alter the sign of the slope. Instead, the transitory shock affects both the

level and the slope of the term-structure of equity premia. Equity compensations are low and

barely flat or slightly increasing in normal and good times. However, compensations increase in

size and become markedly downward-sloping in bad times. Business cycle uncertainty helps

the model to match the conditional dynamics of the term-structure of equity premia and,

hence, to provide further support to the main model mechanism of income insurance. Indeed,

the labor-share, which is decreasing with zt, positively predicts dividend strips volatility and

premium –consistently with the empirical results in Table II and Figure 3. Similarly to the

main model, I compare the model-implied slope of dividend strips premia with actual data

following van Binsbergen and Koijen (2016):

(spot) 15

∫ 5

0(µP − r)(τ)dτ − (µP − r) = 1.4%,

(futures) 15

∫ 5

0[(µP − r)(τ)− (µB − r)(τ)]dτ − [(µP − r)− (µB − r)(10)] = 3.7%.

These values closely resemble their empirical counterparts: 1.2% and 4.1%.

The model of this section leads to time-variation also in the equity premium. Namely, the

40

premium on the market asset is a decreasing function of the transitory shock for γ > ψ > 1.

An usual way to assess the time-varying properties of risk premia is to verify whether the

model captures the pattern of predictability observed in actual data. Thus, I regress long-

horizon excess returns on the log price-dividend ratio. The regression is performed on returns

measured over horizons ranging from 1 to 7 years. In each simulation, the economy is simulated

at the monthly frequency over a 67-year horizon. Panel A of Table X shows that the model

reproduces the negative and statistically significant intertemporal relationship between the

current valuation ratios and future excess returns that we observe in actual data (Cochrane

(2008), Lettau and van Nieuwerburgh (2008), van Binsbergen and Koijen (2010)). Moreover,

the explanatory power and statistical significance increase with the horizon. However, Panel

B shows that the price dividend ratio also predicts consumption growth. The explanatory

power is much smaller but inconsistent with actual data.27

TableXOverall, income insurance, combined with business cycle uncertainty, helps to jointly un-

derstand the timing of dividend risk, the dynamic slope of the term-structure of equity, the

main properties of the market asset and risk-free rate, as well as long-horizon predictability

of equity excess returns.

V. Conclusion

This paper documents that the timing of risk of output, wages and dividends is heterogeneous.

Income insurance from shareholders to workers, which only concerns transitory risk, empiri-

cally and theoretically explains those term-structures and provides a rationale to downward-

sloping dividend risk. Once the resulting dividend dynamics is embedded in an otherwise

standard general equilibrium asset pricing model, the negative slope of dividend risk trans-

mits to equity returns under standard preferences. Thus, income insurance allows to reconcile

in equilibrium traditional asset pricing facts –such as the risk-free rate and equity premium

puzzle, the excess volatility of equity, the increasing real yields on bonds, the long horizon

equity predictability– with the new evidence about the term-structures of both fundamentals

41

and equity and their dynamics. Consistently, labor-share variation hugely forecasts dividend

strips risk, premium and slope.

The model can be extended in a number of directions, such as interests rates and cross-

sectional equity returns. Marfe (2015a, 2015b) empirically and theoretically support the idea

that the term-structure effect of income insurance helps to explain respectively the joint term-

structures of real and nominal interest rates and the dynamics of the value premium.

Appendix

Equilibrium State-Price Density. Under the infinite horizon, the utility process J satisfies the

following Bellman equation: DJ(X,µ, z) + f(Cs, J) = 0, where D denotes the differential operator.

Then we have

0 =JXµX + 12JX,Xσ

2xX

2 + Jµλµ(µ− µ) + 12Jµ,µσ

2µ + Jzλz(z − z) + 1

2Jz,zσ2z + f(Cs, J).

Guess a solution of the form J(X,µ, z) = 11−γX

1−γg(µ, z). The Bellman equation reduces to

0 =µ− 12γσ

2x +

gµgλµ(µ−µ)

1−γ + 12gµ,µg

σ2µ

1−γ + gzgλz(z−z)

1−γ + 12gz,zg

σ2z

1−γ + β1−1/ψ

(g−1/χe(1−1/ψ)(d0+dzz) − 1

).

(A1)

Under limited market participation (Cs,t = Dt) and stochastic differential utility, the pricing kernel

has dynamics given by

dξ0,t = ξ0,tdfCfC

+ ξ0,tfJdt = −r(t)ξ0,t − θx(t)ξ0,tdBx,t − θµ(t)ξ0,tdBµ,t − θz(t)ξ0,tdBz,t, (A2)

where, by use of Ito’s Lemma and Eq. (A1), we get

r(t) = − ∂XfCfC

µX − 12∂X,XfCfC

σ2xX

2 − ∂µfCfC

λµ(µ− µ)− 12∂µ,µfCfC

σ2µ −

∂zfCfC

λz(z − z)− 12∂z,zfCfC

σ2z − fJ ,

θx(t) = − ∂XfCfC

σxX, θµ(t) = −∂µfCfC

σµ, θz(t) = −∂zfCfC

σz,

42

An exact solution for g(µ, z) satisfying Eq. (A1) does not exist for ψ 6= 1. Therefore, I look for

a solution of g(µ, z) around the unconditional mean of the consumption-wealth ratio. Aggregate

wealth is given by

Qs,t = Et[∫ ∞

tξt,uCs,udu

],

and, applying Fubini’s Theorem and taking standard limits, the consumption-wealth ratio satisfies

Cs,tQs,t

= r(t)− 1dtEt

[dQQ

]− 1

dtEt[dξξdQQ

]. (A3)

Guess

Qs,t = Cs,tβ−1(g(µt, zt)e

(γ−1)(d0+dzzt))1/χ

and apply Ito’s Lemma to get dQQ . Then, plug in the wealth dynamics, the risk-free rate and the

pricing kernel into Eq. (A3): after tedious calculus you can recognize that the guess solution is

correct. Notice that the consumption-wealth ratio approaches to β when ψ → 1 as usual.

Denote cq = E[logCs,t − logQs,t], hence, a first-order approximation of the consumption-wealth

ratio around cq produces

Cs,tQs,t

= βg(µt, zt)−1/χe(1−1/ψ)(d0+dzzt) ≈ ecq

(1− cq + log β − 1

χ

(log g(µt, zt) + (γ − 1)(d0 + dzzt)

)).

Using such approximation in the Bellman equation (A1) leads to

0 =µ− 12γσ

2x +

gµgλµ(µ−µ)

1−γ + 12gµ,µg

σ2µ

1−γ + gzgλz(z−z)

1−γ + 12gz,zg

σ2z

1−γ

+ 11−1/ψ

(ecq(

1− cq + log β − 1χ log g(µ, z) + (1− 1/ψ)(d0 + dzz)

)− β

),

which has exponentially affine solution g(µ, z) = eu0+(1−γ)d0+uµµ+(uz+(1−γ)dz)z, where u0, uµ and uz

have explicit solutions and the endogenous constant cq satisfies cq = log β − χ−1(u0 + uµµ + uz z)

43

(recall z = E[zt] = 0). The risk-free rate and the prices of risk take the form:

r0 = 12

(2(βψ2(γ−1)2+ecq((1−ψ)u0+ψ(cq−1−log(β))(γ−1))(ψγ−1))

ψ(1−ψ)(γ−1) + 2z(uz−ψ(uz−dz(γ−1))γ)λzψ(−1+γ)

−2µuµ(ψγ−1)λµψ(γ−1) − γ (1 + γ)σ2

x −(uz−ψ(uz−dz(γ−1))γ)2σ2

zψ2(γ−1)2

− u2µ(ψγ−1)2σ2µ

(ψ−ψγ)2

),

rµ =ψ(γ−1)γ+uµ(ψγ−1)(ecq+λµ)

ψ(γ−1) ,

rz = −ψdz(γ−1)γλz+uz(ψγ−1)(ecq+λz)ψ(γ−1) ,

θx(t) = γσx, θµ(t) =uµ(γ− 1

ψ

)1−γ σµ, θz(t) =

(dzγ + uz(1−ψγ)

ψ(γ−1)

)σz,

and the results in the text easily follow.

PROPOSITION A. The following conditional expectation has exponential affine solution:

Mt,τ (c) = Et[ec0+c1 log ξ0,t+τ+c2xt+τ+c3µt+τ+c4zt+τ ] = ξc10,tXc2t e

`0(τ,c)+`µ(τ,c)µt+`z(τ,c)zt , (A4)

where c = (c0, c1, c2, c3, c4), model parameters are such that the expectation exists finite and `0, `µ

and `z are deterministic functions of time.

Proof: Consider the following conditional expectation:

Mt,τ (c) = Et[ec0+c1 log ξ0,t+τ+c2 logXt+τ+c3µt+τ+c4zt+τ ] (A5)

where c = (c0, c1, c2, c3, c4) is a coefficient vector such that the expectation exists. Guess an expo-

nential affine solution of the kind:

Mt,τ (c) = ec1 log ξ0,t+c2 logXt+`0(τ,c)+`µ(τ,c)µt+`z(τ,c)zt , (A6)

where `0(τ, c), `µ(τ, c) and `z(τ, c) are deterministic functions of time. Feynman-Kac gives that M

44

has to meet the following partial differential equation

0 =Mt −Mξ(r0 + rµµ+ rzz) + 12Mξ,ξ(θx(t)2 + θµ(t)2 + θz(t)

2) +MX(µX) + 12MX,Xσ

2xX

2

+Mµλµ(µ− µ) + 12Mµ,µσ

2µ +Mzλz(z − z) + 1

2Mz,zσ2z −Mξ,Xθx(t)σxX

−Mξ,µθµ(t)σµ −Mξ,zθz(t)σz,

where the arguments have been omitted for ease of notation. Plugging the resulting partial derivatives

from the guess solution into the pde and simplifying gives a linear function of the states µ and z.

Hence, we get three ordinary differential equations for `0(τ, c), `µ(τ, c) and `z(τ, c):

0 = `′0(τ, c)− c1r0 + 12c1(c1 − 1)(θx(t)2 + θµ(t)2 + θz(t)

2) + 12c2(c2 − 1)σ2

x + `µ(τ, c)λµµ

+ 12`µ(τ, c)2σ2

µ + `z(τ, c)λz z + 12`z(τ, c)

2σ2z − c1c2θx(t)σx − c1`µ(τ, c)θµ(t)σµ

− c1`z(τ, c)θz(t)σz

0 = `′µ(τ, c)− c1rµ + c2 − `µ(τ, c)λµ

0 = `′z(τ, c)− c1rz − `z(τ, c)λz

with initial conditions: `0(0, c) = c0, `µ(0, c) = c3 and `z(0, c) = c4. Explicit solutions are avaliable.

Dividends. The conditional moment generating function Dt(τ, n) is a weighted sum of exponential

affine functions. In particular, we have:

Et [Dt+τ ] =Et[Xt+τ

(ezt+τ − αe(1−φ)zt+τ

)]=Mt,τ (c)−Mt,τ (c′),

where c = (0, 0, 1, 0, 1) and c′ = (logα, 0, 1, 0, 1− φ), and

Et[D2t+τ

]=Et

[X2t+τ

(ezt+τ − αe(1−φ)zt+τ

)2]=Mt,τ (c)− 2Mt,τ (c′) +Mt,τ (c′′),

where c = (0, 0, 2, 0, 2), c′ = (logα, 0, 2, 0, 2− φ) and c′′ = (2 logα, 0, 2, 0, 2(1− φ)). With this results

in hand, it is possible to compute the term-structures of the growth rates’ volatility σ2D(t, τ).

Under the log-linearized dynamics in Eq. (10), Dt(τ, n) is exponential affine and obtains as a special

45

case of Mt,τ (c) with c = (nd0, 0, n, 0, ndz). Therefore,

σ2D(t, τ) = vD,τ,xσ

2x + vD,τ,µσ

2µ + vD,τ,zσ

2z ,

where the coefficients are given by

vD,τ,x = 1, vD,τ,µ =4e−λµτ−e−2λµτ+2λµτ−3

2λ3µτ, vD,τ,z = e−λzτ sinh(λzτ)d2z

λzτ.

The moment generating functions and the term-structures of volatility for wages and total consump-

tion are computed in a similar way.

Dividend Strips. The equilibrium price of the market dividend strip with maturity τ of Eq.

(19) obtains as a special case of Mt,τ (c) with c = (d0, 1, 1, 0, dz). Therefore, it is given by Pt,τ =

ξ−10,tMt,τ (c) = Xte

A0(τ)+Aµ(τ)µt+Az(τ)zt with

A0(τ) = `0(τ, c) = 14

(− 4e−τλµ µ(−1+rµ)(1+eτλµ (−1+τλµ))

λµ+ e−2τ(λz+λµ)

λ3zλ

×(−e2τλµ (rz + dzλz)

2λ3µσ

2z + 4eτ(λz+2λµ) (rz + dzλz)λ

(−zλ2

z + σz (θzλz + rzσz))

−e2τλz (−1 + rµ) 2λ3zσ

2µ + 4eτ(2λz+λµ) (−1 + rµ)λ3

zσµ (θµλµ + (−1 + rµ)σµ)

+e2τ(λz+λµ)(λ3µ

(4λ2

z (z (rz + (dz − τrz)λz) + λz (d0 − τ (r0 + θxσx)))− 4θzλz (rz + (dz − τrz)λz)σz

+(d2zλ

2z − 2dzrzλz + r2

z (2τλz − 3))σ2z

)+ 4 (rµ − 1) θµλ

3zλµ (τλµ − 1)σµ + (rµ − 1) 2λ3

z (2τλµ − 3)σ2µ

))),

Aµ(τ) = `µ(τ, c) =1+e−τλµ (−1+rµ)−rµ

λµ,

Az(τ) = `z(τ, c) = −rz+e−τλz (rz+dzλz)λz

,

and A0(0) = d0, Aµ(0) = 0 and Az(0) = dz. Ito’s Lemma gives the dynamics of the market dividend

strip price:

dPt,τ = [·]dt+ ∂xPt,τσxdBx,t + ∂µPt,τσµdBµ,t + ∂zPt,τσzdBz,t.

Therefore the return volatility and premium are given by

σP (t, τ) =P−1t,τ

√(∂xPt,τσx)2 + (∂µPt,τσµ)2 + (∂zPt,τσz)2 =

√σ2x + (Aµ(τ)σµ)2 + (Az(τ)σz)2,

(µP − r)(t, τ) = − 1dt

⟨dξ0,tξ0,t

,dPt,τPt,τ

⟩= θx(t)σx + θµ(t)Aµ(τ)σµ + θz(t)Az(τ)σz.

46

The slopes of the return volatility and premium for the market dividend strip obtain by standard

calculus.

Market Asset and Equity Premium. Under the assumption of limited market participation, the

shareholders act as a representative agent on the financial markets and, hence, the equilibrium price

of the market asset is equal to the shareholders’ wealth. Therefore, using the previous results, the

market asset price can be written as

Pt = Qs,t = Cs,te−cqt = Xte

− log β+u0χ−1+d0+uµχ−1µt+(uzχ−1+dz)zt .

The dynamics of the market asset price obtains by applying Ito’s Lemma to Pt:

dPt = [·]dt+ ∂xPtσxdBx,t + ∂µPtσµdBµ,t + ∂zPtσzdBz,t.

Therefore the return volatility and premium are given by

σP (t) =P−1t

√(∂xPtσx)2 + (∂µPtσµ)2 + (∂zPtσz)2 =

√σ2x + (uµχ−1σµ)2 + ((uzχ−1 + dz)σz)2,

(µP − r)(t) = − 1dt

⟨dξ0,tξ0,t

, dPtPt

⟩= θx(t)σx + θµ(t)uµχ

−1σµ + θz(t)(uzχ

−1 + dz)σz.

Using the definition of the dividend strip price Pt,τ = Et[ξt,t+τDt+τ ] and market asset price Pt =∫∞0 Pt,τdτ , we can write the instantaneous premium of the market return as follows:

(µP − r)(t) = θx(t)∂xPtPtσx + θµ(t)

∂µPtPt

σµ + θz(t)∂zPtPt

σz

= θx(t)∫∞0 ∂xPt,τdτ

Ptσx + θµ(t)

∫∞0 ∂µPt,τdτ

Ptσµ + θz(t)

∫∞0 ∂zPt,τdτ

Ptσz

=

∫ ∞0

P−1t Pt,τ (θx(t)σx + θµ(t)Aµ(τ)σµ + θz(t)Az(τ)σz) dτ

=

∫ ∞0

Π(t, τ)dτ,

and Π(t, τ) easily follows using the previous results.

Bond and Equity Yields. The equilibrium price of the zero-coupon bond with maturity τ obtains

as a special case of Mt,τ (c) with c = (0, 1, 0, 0, 0). Then, it is given by Bt,τ = ξ−10,tMt,τ (c). And the

premium is: (µB − r)(t, τ) = − 1dt

⟨dξ0,tξ0,t

,dBt,τBt,τ

⟩.

47

Given p(t, τ) = log(Dt/Pt,τ )/τ and gD(t, τ) = log(Dt(τ, 1)/Dt)/τ , it is easy to verify that the premium

on the equity yield, %(t, τ) = p(t, τ)− ε(t, τ) + gD(t, τ) in Eq. (34), is state-independent.

48

REFERENCES

Ai, Hengjie, Mariano Massimiliano Croce, Anthony M. Diercks, and Kai Li, 2015, News shocks and

production-based term structure of equity returns, Unpublished manuscript.

Aıt-Sahalia, Yacine, Mustafa Karaman, and Loriano Mancini, 2015, The term structure of variance

swaps and risk premia, Unpublished manuscript.

Alvarez, Fernando, and Urban J. Jermann, 2005, Using Asset Prices to Measure the Persistence of

the Marginal Utility of Wealth, Econometrica 73, 1977–2016.

Andries, Marianne, Thomas M. Eisenbach, and Martin C. Schmalz, 2014, Asset pricing with horizon-

dependent risk aversion, Unpublished manuscript.

Azariadis, Costas, 1978, Escalator clauses and the allocation of cyclical risks, Journal of Economic

Theory 18, 119–155.

Baily, Martin N., 1974, Wages and employment under uncertain demand, The Review of Economic

Studies 41, pp. 37–50.

Bansal, Ravi, Dana Kiku, and Amir Yaron, 2010, Long run risks, the macroeconomy, and asset

prices, American Economic Review 100, 542–46.

Bansal, Ravi, and Amir Yaron, 2004, Risks for the long run: A potential resolution of asset pricing

puzzles, Journal of Finance 59, 1481–1509.

Barberis, Nicholas, Ming Huang, and Tano Santos, 2001, Prospect theory and asset prices, The

Quarterly Journal of Economics 116, 1–53.

Beeler, Jason, and John Y. Campbell, 2012, The long-run risks model and aggregate asset prices:

An empirical assessment, Critical Finance Review 1, 141–182.

Belo, Frederico, Pierre Collin-Dufresne, and Robert S. Goldstein, 2015, Endogenous dividend dy-

namics and the term structure of dividend strips, Journal of Finance 70, 1115–1160.

Benzoni, Luca, Pierre Collin-Dufresne, and Robert S. Goldstein, 2011, Explaining asset pricing

puzzles associated with the 1987 market crash, Journal of Financial Economics 101, 552 – 573.

49

Berk, Jonathan B., and Johan Walden, 2013, Limited capital market participation and human capital

risk, Review of Asset Pricing Studies 3, 1–37.

Berrada, Tony, Jerome Detemple, and Marcel Rindisbacher, 2013, Asset pricing with regime-

dependent preferences and learning, Unpublished manuscript.

Boguth, Oliver, Murray Carlson, Adlai Fisher, and Mikhail Simutin, 2012, Dividend strips and

the term structure of equity risk premia: A case study of the limits of arbitrage, Unpublished

manuscript.

Boldrin, Michele, and Michael Horvath, 1995, Labor contracts and business cycles, Journal of Political

Economy 103, 972–1004.

Campbell, John Y., and John H. Cochrane, 1999, By force of habit: A consumption based explanation

of aggregate stock market behavior, Journal of Political Economy 107, pp. 205–251.

Cochrane, John H., 2008, The dog that did not bark: A defense of return predictability, Review of

Financial Studies 21, 1533–1575.

Constantinides, George M., and Anisha Ghosh, 2011, Asset pricing tests with long-run risks in

consumption growth, Review of Asset Pricing Studies 1, 96–136.

Croce, Mariano M., Martin Lettau, and Sydney C. Ludvigson, 2015, Investor information, long-run

risk, and the term structure of equity, Review of Financial Studies (forthcoming).

Curatola, Giuliano, 2015, Loss aversion, habit formation and the term structures of equity and

interest rates, Journal of Economic Dynamics and Control 53, 103–122.

Danthine, Jean-Pierre, and John B. Donaldson, 1992, Risk sharing in the business cycle, European

Economic Review 36, 468–475.

Danthine, Jean-Pierre, and John B. Donaldson, 2002, Labour relations and asset returns, Review of

Economic Studies 69, 41–64.

Danthine, Jean-Pierre, John B Donaldson, and Paolo Siconolfi, 2005, Distribution Risk and Equity

Returns, CEPR Discussion Papers 5425, C.E.P.R. Discussion Papers.

50

Dechow, Patricia M., Richard G. Sloan, and Mark T. Soliman, 2004, Implied equity duration: A new

measure of equity risk, Review of Accounting Studies 9, 197–228.

Dew-Becker, Ian, 2016, How risky is consumption in the long-run? benchmark estimates from a

robust estimator, Review of Financial Studies .

Donangelo, Andres, 2014, Labor mobility: Implications for asset pricing, Journal of Finance 69,

1321–1346.

Duffie, Darrell, and Larry G. Epstein, 1992, Stochastic differential utility, Econometrica 60, 353–94.

Ellul, Andrew, Marco Pagano, and Fabiano Schivardi, 2014, Employment and Wage Insurance within

Firms: Worldwide Evidence, CSEF Working Papers 369, Centre for Studies in Economics and

Finance (CSEF), University of Naples, Italy.

Epstein, Larry G., and Stanley E. Zin, 1989, Substitution, risk aversion, and the temporal behavior

of consumption and asset returns: A theoretical framework, Econometrica 57, 937–69.

Favilukis, Jack, and Xiaoji Lin, 2015, Wage rigidity: A quantitative solution to several asset pricing

puzzles, Review of Financial Studies (forthcoming).

Gabaix, Xavier, 2012, Variable rare disasters: An exactly solved framework for ten puzzles in macro-

finance, The Quarterly Journal of Economics 127, 645–700.

Gamber, Edward N, 1988, Long-term risk-sharing wage contracts in an economy subject to permanent

and temporary shocks, Journal of Labor Economics 6, 83–99.

Gertler, Mark, and Antonella Trigari, 2009, Unemployment Fluctuations with Staggered Nash Wage

Bargaining, Journal of Political Economy 117, 38–86.

Gomme, Paul, and Jeremy Greenwood, 1995, On the cyclical allocation of risk, Journal of Economic

Dynamics and Control 19, 91–124.

Greenwald, Daniel L., Martin Lettau, and Sydney C. Ludvigson, 2014, Origins of stock market

fluctuations, Unpublished manuscript, National Bureau of Economic Research.

51

Guiso, Luigi, Luigi Pistaferri, and Fabiano Schivardi, 2005, Insurance within the firm, Journal of

Political Economy 113, 1054–1087.

Hansen, Lars Peter, John C. Heaton, and Nan Li, 2008, Consumption Strikes Back? Measuring

Long-Run Risk, Journal of Political Economy 116, 260–302.

Hansen, Lars Peter, and Jos A. Scheinkman, 2009, Long-Term Risk: An Operator Approach, Econo-

metrica 77, 177–234.

Hasler, Michael, and Roberto Marfe, 2015, Disaster recovery and the term-structure of dividend

strips, Journal of Financial Economics (forthcoming).

Khapko, Mariana, 2014, Asset pricing with dynamically inconsistent agents, Unpublished manuscript.

Knight, Frank H., 1921, Risk, Uncertainty and Profit (Houghton Mifflin).

Kogan, Leonid, and Dimitris Papanikolaou, 2015, Firm characteristics and stock returns: The role

of investment-specific shocks, Review of Financial Studies .

Koijen, Ralph, Hanno Lusting, and Stijn van Nieuwerburgh, 2014, The cross-section and time-series

of stock and bond returns, Unpublished manuscript.

Kreps, David M., and Evan L. Porteus, 1979, Temporal von neumann-morgenstern and induced

preferences, Journal of Economic Theory 20, 81–109.

Kuehn, Lars-Alexander, Nicolas Petrosky-Nadeau, and Lu Zhang, 2012, An equilibrium asset pricing

model with labor market search, Technical report.

Larrain, Borja, and Motohiro Yogo, 2008, Does firm value move too much to be justified by subsequent

changes in cash flow?, Journal of Financial Economics 87, 200 – 226.

Lettau, Martin, and Sydney C. Ludvigson, 2005, Expected returns and expected dividend growth,

Journal of Financial Economics 76, 583–626.

Lettau, Martin, and Sydney C. Ludvigson, 2014, Shocks and Crashes, NBER Macroeconomics Annual

28, 293 – 354.

52

Lettau, Martin, and Stijn van Nieuwerburgh, 2008, Reconciling the return predictability evidence,

Review of Financial Studies 21, 1607–1652.

Lettau, Martin, and Jessica A. Wachter, 2007, Why Is Long-Horizon Equity Less Risky? A Duration-

Based Explanation of the Value Premium, Journal of Finance 62, 55–92.

Lettau, Martin, and Jessica A. Wachter, 2011, The term structures of equity and interest rates,

Journal of Financial Economics 101, 90–113.

Longstaff, Francis A., and Monika Piazzesi, 2004, Corporate earnings and the equity premium,

Journal of Financial Economics 74, 401–421.

Lopez, Pierlauro, 2014, The term structure of the welfare cost of uncertainty, Unpublished

manuscript.

Lopez, Pierlauro, David Lopez-Salido, and Francisco Vazquez-Grande, 2015, Nominal rigidities and

the term structures of equity and bond returns, Unpublished manuscript.

Marfe, Roberto, 2014, Demand shocks, timing preferences and the equilibrium term structures,

Unpublished manuscript.

Marfe, Roberto, 2015a, Labor rigidity and the dynamics of the value premium, Unpublished

manuscript.

Marfe, Roberto, 2015b, Labor rigidity, inflation risk and bond returns, Unpublished manuscript.

Marfe, Roberto, 2016, Corporate fraction and the equilibrium term structure of equity risk, Review

of Finance 20, 855–905.

Menzio, Guido, 2005, High-frequency wage rigidity, Unpublished manuscript.

Rıos-Rull, Jose-Vıctor, and Raul Santaeulalia-Llopis, 2010, Redistributive shocks and productivity

shocks, Journal of Monetary Economics 57, 931–948.

Santos, Tano, and Pietro Veronesi, 2006, Labor income and predictable stock returns, Review of

Financial Studies 19, 1–44.

53

Schulz, Florian, 2016, On the timing and pricing of dividends: Comment, American Economic Review

(forthcoming).

Shimer, Robert, 2005, The cyclical behavior of equilibrium unemployment and vacancies, American

Economic Review 95, 25–49.

van Binsbergen, Jules, Michael Brandt, and Ralph Koijen, 2012, On the timing and pricing of

dividends, American Economic Review 102, 1596–1618.

van Binsbergen, Jules, and Ralph Koijen, 2012b, A note on “Dividend strips and the term structure

of equity risk premia: A case study of the limits of arbitrage”, Unpublished manuscript.

van Binsbergen, Jules H., Wouter H. Hueskes, Ralph Koijen, and Evert B. Vrugt, 2013, Equity yields,

Journal of Financial Economics 110, 503–519.

van Binsbergen, Jules H., and Ralph Koijen, 2016, The term structure of returns: Facts and theory,

Journal of Financial Economics (forthcoming).

van Binsbergen, Jules H., and Ralph S. J. Koijen, 2010, Predictive regressions: A present-value

approach, The Journal of Finance 65, 1439–1471.

Weber, Michael, 2016, The term structure of equity returns: Risk or mispricing?, Unpublished

manuscript.

Weil, Philippe, 1989, The equity premium puzzle and the risk-free rate puzzle, Journal of Monetary

Economics 24, 401–421.

54

Notes

1Examples are the seminal works by Campbell and Cochrane (1999), Barberis, Huang, and Santos (2001),

Bansal and Yaron (2004) and Gabaix (2012), among others.

2The idea that the very role of the firm is that of insurance provider has a long tradition since Knight (1921),

Baily (1974), Azariadis (1978), Boldrin and Horvath (1995) and Gomme and Greenwood (1995). They suggest

that workers’ remuneration is partially fixed in advance and, hence, shareholders bear most of aggregate risk

but, in exchange of income insurance, gain flexibility in labor supply. Recently, Guiso et al. (2005), Shimer

(2005) and Rıos-Rull and Santaeulalia-Llopis (2010) provide empirical support.

3Beyond the focus on the term-structures, the model differs from Greenwald et al. (2014) because: (i) the

labor-share is not an exogenous shock but derives from income insurance and has counter-cyclical dynamics as

in Rıos-Rull and Santaeulalia-Llopis (2010); (ii) the model does not account for preference shocks –which help

to generate a-cyclical variation in equity returns– since the scope of the paper concerns the term-structure

effect of income insurance; (iii) the model state-price density is fully endogenous, whereas Greenwald et al.

(2014) specify an exogenous and constant risk-free rate; (iv) the model calibration exploits information from

the timing of macroeconomic risk, whereas it is different in Greenwald et al. (2014).

4Marfe (2016) considers the term-structure of equity in the context of a long-run risk model: consumption

and dividends co-integration leads to decreasing equity volatility but increasing premia. Berrada, Detemple,

and Rindisbacher (2013), Andries, Eisenbach, and Schmalz (2014), Curatola (2015), Khapko (2014) and Marfe

(2014) focus on non-standard beliefs and state-dependent risk preferences. Lopez, Lopez-Salido, and Vazquez-

Grande (2015) consider nonlinear habit and nominal rigidities. Lopez (2014) investigates welfare costs.

5Even if the model is silent on production and workers saving decisions, Danthine, Donaldson, and Siconolfi

(2005) show that a similar form of distributional risk improves the asset pricing implications of a standard

real business cycle model without deteriorating business cycle predictions.

6The baseline model calibration overestimates the 20-years maturity real yield from TIPs data by about

200 basis points.

7Alvarez and Jermann (2005) show that the fraction of state-price density variation driven by its martingale

component is increasing in the market return in excess of the long-term real bond yield. The low historical

level of the latter implies a prominent role of the martingale component. See also Hansen and Scheinkman

(2009) and Hansen, Heaton, and Li (2008). Here, while the model state-price density inherits decreasing

variance-ratios from dividends, its long-run volatility is still sizeable –about 32% (see the Online Appendix

A). This is due to the joint effect of consumption and dividends co-integration, time-varying long-run growth,

and preferences for the early resolution of uncertainty.

8In the Online Appendix A, I document that payout to shareholders from the aggregate economy (NIPA

55

tables) and from NYSE, NASDAQ, and Amex firms (CRSP) feature markedly downward-sloping VR’s. More-

over, the VR’s of aggregate consumption are about flat and close to those of value added (see Figure 4). This

result is consistent with the recent work by Dew-Becker (2016): robust estimators of long-run consumption

volatility imply an approximatively flat term-structure of consumption VR’s.

9The Online Appendix A shows that the term-structure of risk of value added minus investments is almost

identical to that of value added. Thus, investments do not induce a term-structure effect.

10Conditional volatilities are estimated with an ARMA(1,1)-Garch(1,1) model and moving averages are

computed with a rolling window of ±5 years. Estimation details are reported in the Online Appendix A.

11For small τ , the effect for wages is less prominent than the effect for dividends because the wages are

mainly driven by permanent shocks. This is consistent with the mechanism of income insurance. Indeed, for

τ large enough, VR’s are more informative about the slope of the term-structure and one can observe a highly

significant positive relation with the labor-share.

12The Online Appendix A documents that the results of Table I are robust to a weak time-trend in the

labor-share.

13Monthly dividend strip excess return and AR(1)-GARCH(1,1) return volatility are available from Ralph

Koijen’s webpage and correspond to ‘Strategy 1’ in van Binsbergen et al. (2012). Namely, returns represent

a one-month long position in short-term dividend strips (average maturity is about 1.6 years).

14In the Online Appendix A, I find that ruling out a weak time trend from the labor-share leads to an even

stronger relationship.

15Consistently with the former analysis of the term-structure effect of income insurance, I also verify that

the time-series of the variance ratios of wages minus those of dividends significantly and positively predicts

dividend strips volatility. Newey-West t-statistics and adjusted-R2 are respectively 3.00 and 41% (one-quarter

ahead), 3.99 and 51% (one-year ahead) and 5.21 and 62% (two-years ahead). The coefficient remains positive

and highly significant in all of the horse race regressions and the adjusted-R2 do not increase substantially.

Details in the Online Appendix A.

16The Online Appendix A documents that the labor-share significantly forecasts dividend strip return volatil-

ity, excess return over the risk-free rate and over the market return 1-quarter, 1-year and 2-years ahead both

in multivariate and in horse race regressions.

17I pursue additional robustness checks in the Online Appendix A. The financial leverage ratio leads to

additional explanatory power for dividend growth but it does not help to predict consumption growth. Vice-

versa, the investment-share does not help to forecast dividend growth but it increases consumption growth

predictability.

18The Online Appendix A reports the term-structures of risk of total payout by Larrain and Yogo (2008)

56

(net interests, net debt issuances, net cash dividends and net equity issuances), shareholders’ payout (net cash

dividends and net equity issuances) and net cash dividends from the non-financial corporate sector on the

sample 1927-2004. The variance ratios of all these variables are markedly downward-sloping, similar to each

other and similar to the variance ratios of the remainder of output minus wages. Thus, the payout policy

likely does not induce a term-structure effect.

19The endogenous determination of market participants goes beyond the scope of this paper. The assump-

tion of limited market participation allows for tractability and for comparability with most of endowment

economy equilibrium models. Recently, Berk and Walden (2013) show that labor markets provide risk-sharing

to workers, such that their consumption endogenously equals their wages and limited market participation

obtains.

20Under CRRA preferences (ψ → 1/γ), the optimality condition for the shareholders takes the usual power

form: IC [ξ0,t] = (ξ0,teβt)−1/γ and the term capturing the distributional risk is unchanged. Instead, for φ→ 0,

the state-price density reduces to ξ0,t = e−βt((1− α)Ct)−γ , as in a Lucas economy.

21A similar result obtains measuring the slopes by means of the term-spreads σ2P (t,∞) − σ2

P (t, 0) and

(µP –r)(t,∞)−(µP –r)(t, 0). Negative spreads obtain if the leverage effect of dividends due to income insurance

is large enough.

22Standardizing Π(t, τ) by (µP − r)(t) allows to compare the timing decomposition of the equity premium

among models and parameter settings even if they produce different magnitudes for the premium.

23Monotone downward-sloping real yields obtain in most of long-run risk models. Here, for income insurance

large enough, real yields are upward-sloping consistently with the slope of TIPs data.

24The term-structures of premium, volatility and Sharpe ratio of dividend strip futures returns are reported

in the Online Appendix A.

25The model-implied equity duration, i.e. P−1t

∫∞0τPt,τdτ , is about 35.6 years which is consistent with the

estimates by Dechow, Sloan, and Soliman (2004).

26See Boguth, Carlson, Fisher, and Simutin (2012), van Binsbergen and Koijen (2012b), Weber (2016) and

Schulz (2016).

27Note that the model only accounts for priced shocks. Shocks to dividends orthogonal to consumption could

likely reduce the huge explanatory power for excess returns and rule out consumption growth predictability.

Such a shock to dividends has not been considered for the sake of simplicity and exposition and Table V

documents that it is not relevant to the scope of the paper.

57

Table I: The Term-Structure Effect of Income Insurance

The table reports the estimates of the regressions:

VRW − VRD

∣∣t,τ

= a + bwd W/Yt + byVRY

∣∣t,τ

+ b′c controlst + εt,

VRW

∣∣t,τ

= a + bw W/Yt + byVRY

∣∣t,τ

+ b′c controlst + εt,

VRD

∣∣t,τ

= a + bd W/Yt + byVRY

∣∣t,τ

+ b′c controlst + εt,

where the dependent variables are either the variance ratios of wages, those of dividends or

their difference computed at time t with horizon τ ranging from 2 to 7 years; the independent

variables are the time t wages over value added, the time t variance ratios of value added with

horizon τ and the time t orthogonalized investment over value added and financial leverage

as controls. Data are from the non-financial corporate sector (Flow of Funds) on the sample

1951:4-2015:1 at quarterly frequency. The time-series of variance ratios are computed through

a rolling window of 10 years centered at t. Newey-West corrected t-statistics are reported in

parenthesis. The symbols ∗, ∗∗, and ∗∗∗ denote statistical significance at the 10%, 5%, and 1%

levels.

58

Table I: Continued

Panel A – Dependent variable: VRW −VRD|t,τ

τ 2 3 4 5 6 7

W/Y 6.35∗∗∗ 9.08∗∗∗ 10.63∗∗∗ 11.85∗∗∗ 14.41∗∗∗ 16.98∗∗∗

t-stat (5.50) (5.97) (4.55) (3.74) (3.45) (3.17)

adj-R2 0.42 0.53 0.55 0.57 0.58 0.56

W/Y 6.35∗∗∗ 9.08∗∗∗ 10.63∗∗∗ 11.85∗∗∗ 14.41∗∗∗ 16.98∗∗∗

t-stat (5.77) (5.83) (4.55) (3.91) (3.74) (3.66)

I/Y⊥ -1.50 0.54 3.85 8.72∗∗ 12.81∗∗∗ 17.25∗∗∗

t-stat (-0.87) (0.23) (1.18) (2.17) (2.62) (2.94)

Fin Lev⊥ 0.62∗∗ 0.75∗ 0.61 0.18 0.16 -0.07

t-stat (2.00) ( 1.86) (1.11) (0.26) (0.20) (-0.07)

adj-R2 0.46 0.55 0.58 0.63 0.66 0.67

Panel B – Dependent variable: VRW |t,τ

τ 2 3 4 5 6 7

W/Y 1.38 2.97 4.55∗ 6.51∗∗ 8.86∗∗ 11.43∗∗

t-stat (1.26) (1.58) (1.78) (1.98) (2.06) (2.14)

adj-R2 0.42 0.43 0.48 0.51 0.51 0.50

W/Y 1.38 2.97 4.55∗ 6.51∗∗ 8.86∗∗ 11.43∗∗

t-stat (1.30) (1.63) (1.80) (2.03) (2.15) (2.34)

I/Y⊥ -1.95 -0.15 2.02 5.82 9.9∗∗ 13.76∗∗

t-stat (-1.46) (-0.07) (0.69) (1.62) (2.17) (2.45)

Fin Lev⊥ -0.03 -0.28 -0.34 -0.58 -0.81 -1.05

t-stat (-0.16) (-0.78) (-0.68) (-0.95) (-1.07) (-1.16)

adj-R2 0.46 0.43 0.49 0.54 0.56 0.58

59

Table I: Continued

Panel C – Dependent variable: VRD|t,τ

τ 2 3 4 5 6 7

W/Y -4.97∗∗∗ -6.11∗∗∗ -6.08∗∗∗ -5.34∗∗∗ -5.55∗∗∗ -5.55∗∗∗

t-stat (-4.39) (-5.11) (-5.41) (-4.67) (-4.53) (-4.21)

adj-R2 0.32 0.31 0.31 0.29 0.33 0.34

W/Y -4.97∗∗∗ -6.11∗∗∗ -6.08∗∗∗ -5.34∗∗∗ -5.55∗∗∗ -5.55∗∗∗

t-stat (-5.31) (-6.83) (-7.22) (-6.04) (-6.05) (-5.99)

I/Y⊥ -0.45 -0.69 -1.83 -2.9∗∗ -2.91∗∗ -3.49∗∗

t-stat (-0.37) (-0.46) (-1.23) (-1.98) (-2.21) (-2.79)

Fin Lev⊥ -0.66∗∗∗ -1.02∗∗∗ -0.95∗∗∗ -0.76∗∗∗ -0.98∗∗∗ -0.98∗∗∗

t-stat (-2.98) (-4.95) (-4.85) (-3.50) (-4.42) (-3.91)

adj-R2 0.42 0.47 0.50 0.48 0.57 0.59

60

Table II: Income Insurance and the Term-Structure of Equity

The table reports the coefficient, Newey-West t-statistic and adjusted-R2 of the regressions:

short-term equity risk: volstript = a + brisk W/Yt−1 + εt,

short-term equity premium: rstript − rrisk-freet = a + bpremium W/Yt−1 + εt,

short-term equity slope: rstript − rmktt = a + bslope W/Yt−1 + εt,

where the dependent variables are either the time t dividend strip return volatility or the excess

return over the risk-free rate or the excess return over the market return; the independent

variable is the time t − 1 wages over value added from the non-financial corporate sector.

Data are quarterly on the sample 1996:4-2010:1. Symbols ∗, ∗∗, and ∗∗∗ denote statistical

significance at 10%, 5%, and 1% levels.

Risk Premium Slope

1-quarter

ahead

1-year

ahead

2-years

ahead

1-quarter

ahead

1-year

ahead

2-years

ahead

1-quarter

ahead

1-year

ahead

2-years

ahead

(1) (2) (3) (4) (5) (6) (7) (8) (9)

W/Yt−1 1.10∗∗∗ 1.04∗∗∗ 0.88∗∗∗ 0.43∗ 0.39∗∗∗ 0.37∗∗∗ 0.58∗ 0.53∗∗ 0.43∗∗

t-stat (5.88) (6.03) (5.37) (1.99) (3.02) (6.26) (1.98) (2.58) (2.43)

adj-R2 0.72 0.74 0.62 0.08 0.31 0.64 0.10 0.26 0.28

61

Table III: Income Insurance and the Term-Structure of Equity

The table reports the coefficients, Newey-West t-statistics and adjusted-R2 of the regressions:

short-term equity risk: volstript = a + brisk W/Yt−1 + b′c controlst + εt,

short-term equity premium: rstript − rrisk-freet = a + bpremium W/Yt−1 + b′c controlst + εt,

short-term equity slope: rstript − rmktt = a + bslope W/Yt−1 + b′c controlst + εt,

where the dependent variables are either time t dividend strip return volatility or dividend strip

excess return over the risk-free rate or over the market return; the independent variables are

the time t−1 wages over value added and the time t controls: Fama and French factors (excess

market return, SMB, HML), market volatility, price-earnings ratio, risk-free rate, cay and the

financial leverage and investment over value added from the non-financial corporate sector.

Data are quarterly on the sample 1996:4-2010:1. Symbols ∗, ∗∗, and ∗∗∗ denote statistical

significance at 10%, 5%, and 1% levels.

62

Table III: Continued

Panel A – Dependent variable: 1 year ahead dividend strip return volatility

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

W/Yt−1 1.04∗∗∗ 1.01∗∗∗ 1.05∗∗∗ 1.01∗∗∗ 0.92∗∗∗ 1.03∗∗∗ 1.06∗∗∗ 1.00∗∗∗ 1.03∗∗∗ 1.04∗∗∗

t-stat (6.03) (6.71) (5.89) (7.12) (4.79) (6.08) (6.76) (5.58) (5.97) (9.08)

MKTt -0.19∗∗

t-stat (-2.31)

SMBt -0.08

t-stat (-0.83)

HMLt 0.14

t-stat (0.99)

VOLt 0.34∗

t-stat (1.86)

P/Et 0.00

t-stat (0.15)

Rft 3.42

t-stat (1.31)

CAYt 0.14

t-stat (0.95)

FLt 0.02

t-stat (0.36)

I/Yt 0.72∗∗∗

t-stat (3.84)

adj-R2 0.74 0.78 0.74 0.75 0.76 0.73 0.77 0.74 0.74 0.84

63

Table III: Continued

Panel B – Dependent variable: 1 year ahead dividend strip return over risk-free rate

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

W/Yt−1 0.39∗∗∗ 0.38∗∗∗ 0.41∗∗∗ 0.35∗∗∗ 0.38∗∗ 0.43∗∗∗ 0.39∗∗∗ 0.46∗∗∗ 0.38∗∗∗ 0.39∗∗∗

t-stat (3.02) (3.11) (3.18) (3.64) (2.65) (3.72) (2.94) (3.55) (2.85) (2.96)

MKTt -0.05

t-stat (-0.78)

SMBt -0.12

t-stat (-0.82)

HMLt 0.20

t-stat (1.65)

VOLt 0.03

t-stat (0.26)

P/Et -0.01

t-stat (-0.85)

Rft 0.56

t-stat (0.29)

CAYt -0.27∗∗

t-stat (-2.34)

FLt 0.06

t-stat (1.40)

I/Yt -0.04

t-stat (-0.16)

adj-R2 0.31 0.31 0.32 0.39 0.30 0.31 0.30 0.33 0.33 0.30

64

Table III: Continued

Panel C – Dependent variable: 1 year ahead dividend strip return over market return

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

W/Yt−1 0.53∗∗ 0.49∗∗∗ 0.55∗∗ 0.45∗∗∗ 0.60∗∗ 0.58∗∗∗ 0.53∗∗ 0.51∗ 0.52∗∗ 0.53∗∗∗

t-stat (2.58) (2.88) (2.52) (3.11) (2.48) (2.99) (2.46) (2.06) (2.46) (3.16)

MKTt -0.30∗∗∗

t-stat (-3.39)

SMBt -0.10

t-stat (-0.43)

HMLt 0.42∗∗∗

t-stat (2.80)

VOLt -0.20

t-stat (-0.79)

P/Et -0.01

t-stat (-0.56)

Rft 0.14

t-stat (0.04)

CAYt 0.09

t-stat (0.29)

FLt 0.11∗∗

t-stat (2.07)

I/Yt 0.53

t-stat (1.42)

adj-R2 0.26 0.40 0.25 0.43 0.25 0.25 0.24 0.24 0.30 0.32

65

Table IV: Income Insurance and Expected Growth.

The table reports the coefficient, Newey-West and Hansen-Hodrick t-statistics (four lags) and

adjusted-R2 of the regressions of cumulative dividend growth rates and consumption growth

rates over the horizons of 1, 5, 10 and 15 years on the current labor-share:

∑n

i=1∆xt+i = a+ bW/Yt + εt, ∆xt = {log Dt

Dt−1, log Ct

Ct−1}.

Consumption is NIPA non-durable goods and services; dividends and labor-share are from the

non-financial corporate sector on the sample 1946-2013 at yearly frequency. The symbols ∗,

∗∗, and ∗∗∗ denote statistical significance at the 10%, 5%, and 1% levels (Newey-West standard

errors).

Panel A – Dividends Panel B – Consumption

Horizon (years) 1 5 10 15 1 5 10 15

W/Y -0.12 2.69∗∗ 4.57∗∗∗ 6.93∗∗∗ 0.17∗ 0.51∗∗ 1.01∗∗∗ 1.44∗∗∗

NW t-stat (-0.24) (2.54) (2.76) (4.08) (2.09) (2.57) (4.92) (6.08)

HH t-stat (-0.49) (3.33) (2.37) (3.64) (2.13) (2.49) (5.22) (6.00)

adj-R2 0.00 0.10 0.24 0.37 0.06 0.12 0.30 0.36

66

Table V: Income Insurance and Other Sources of Transitory Dividend Risk

The table reports coefficients, Newey-West t-statistics and adjusted-R2 of the regressions:

dividend slope: VRD|t,τ = a+ bwW/Yt + bdD/Y⊥t + byVRY |t,τ + εt,

short-term equity risk: volstript = a+ bwW/Yt−1 + bdD/Y⊥t−1 + εt,

short-term equity premium: rstript − rrisk-freet = a+ bwW/Yt−1 + bdD/Y⊥t−1 + εt,

short-term equity slope: rstript − rmktt = a+ bwW/Yt−1 + bdD/Y⊥t−1 + εt.

In regression (1), the time t variance ratio of dividends with horizon τ is regressed on the time

t labor-share, the time t orthogonalized dividend-share and the time t variance ratio of value

added with horizon τ . Data are from the non-financial corporate sector (Flow of Funds) on the

sample 1951:4-2015:1 at quarterly frequency. The time-series of variance ratios are computed

through a rolling window of 10 years centered at t. In regressions (2)-(3)-(4), either the time

t dividend strip return volatility or dividend strip excess return over the risk-free rate or over

the market return are regressed on the time t−1 labor-share and the time t−1 orthogonalized

dividend-share. Data are quarterly on the sample 1996:4-2010:1. The symbols ∗, ∗∗, and ∗∗∗

denote statistical significance at the 10%, 5%, and 1% levels.

67

Table V: Continued

Panel A – Dividend Variance Ratios

τ 2 3 4 5 6 7

W/Y -4.97∗∗∗ -6.11∗∗∗ -6.08∗∗∗ -5.34∗∗∗ -5.55∗∗∗ -5.55∗∗∗

t-stat (-4.42) (-4.97) (-5.23) (-4.64) (-4.58) (-4.30)

D/Y⊥ 2.14 2.11 1.70 0.28 -0.31 -0.88

t-stat (1.58) (1.07) (0.95) (0.17) (-0.21) (-0.60)

adj-R2 0.34 0.32 0.32 0.28 0.33 0.34

Panel B – Equity Risk, Premium and Slope

Risk Premium Slope

1-quarter

ahead

1-year

ahead

2-years

ahead

1-quarter

ahead

1-year

ahead

2-years

ahead

1-quarter

ahead

1-year

ahead

2-years

ahead

W/Yt−1 1.10∗∗∗ 1.04∗∗∗ 0.88∗∗∗ 0.43∗ 0.39∗∗∗ 0.37∗∗∗ 0.58∗ 0.53∗∗ 0.43∗∗

t-stat (5.96) (6.14) (5.42) (1.98) (3.01) (6.42) (2.00) (2.66) (2.55)

D/Y⊥t−1 0.12 0.14 0.04 -0.02 0.02 -0.06∗∗ 0.32 0.39 0.24

t-stat (1.15) (1.28) (0.31) (-0.15) (0.20) (-2.45) (-1.12) (1.45) (1.32)

adj-R2 0.72 0.74 0.62 0.06 0.30 0.64 0.10 0.30 0.30

68

Table VI: Calibration – Model Parameters

Symbol Value

volatility of permanent shock σx 0.005

steady-state long-run growth µ 0.020

reversion of long-run growth λµ 0.857

volatility of long-run growth σµ 0.018

reversion of transitory shock λz 0.198

volatility of transitory shock σz 0.026

wages over wages plus dividends α 0.942

income insurance φ 0.396

69

Table VII: Calibration – Cash-Flows Moments

Symbol Data Model

long-run growth gC 0.020 0.020

one year consumption volatility σC(1) 0.025 0.025

one year dividends volatility σD(1) 0.174 0.174

dividend-share average (D/(D+W)) δ 0.060 0.058

dividend-share volatility σδ 0.016 0.015

Variance ratios of consumption V RC(τ)

τ 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Data 0.98 0.95 0.90 0.85 0.81 0.77 0.74 0.74 0.75 0.77 0.80 0.84 0.88 0.90

Model 0.99 0.98 0.97 0.95 0.93 0.91 0.89 0.88 0.87 0.86 0.85 0.84 0.83 0.83

Variance ratios of wages V RW (τ)

τ 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Data 1.20 1.33 1.42 1.47 1.51 1.54 1.55 1.57 1.58 1.59 1.60 1.60 1.58 1.54

Model 1.18 1.32 1.41 1.47 1.51 1.54 1.56 1.57 1.58 1.59 1.59 1.60 1.60 1.61

Variance ratios of dividends V RD(τ)

τ 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Data 0.62 0.40 0.28 0.22 0.20 0.20 0.20 0.20 0.19 0.18 0.16 0.14 0.14 0.19

Model 0.84 0.71 0.61 0.54 0.47 0.42 0.38 0.34 0.31 0.29 0.26 0.25 0.23 0.22

70

Table VIII: Income Insurance and Expected Growth.

The table reports the coefficient, Newey-West t-statistics and adjusted-R2 of the regressions

of cumulative dividend growth rates and consumption growth rates over the horizons of 1, 5,

10 and 15 years on the current labor-share:

∑n

i=1∆xt+i = a+ bW/Yt + εt, ∆xt = {log Dt

Dt−1, log Ct

Ct−1}.

One thousand simulations are run at the monthly frequency over a 67-year horizon. The

symbols ∗, ∗∗, and ∗∗∗ denote statistical significance at the 10%, 5%, and 1% levels.

Panel A – Dividends Panel B – Consumption

Horizon (years) 1 5 10 15 1 5 10 15

W/Y 4.63∗∗∗ 15.32∗∗∗ 19.65∗∗∗ 20.58∗∗∗ 0.64∗∗ 2.01∗∗ 2.67∗ 2.93

t-stat (2.66) (3.43) (4.00) (3.89) ( 2.14) (2.09) (1.85) (1.50)

adj-R2 0.10 0.37 0.47 0.49 0.09 0.22 0.26 0.26

71

Table IX: Return Moments

Data

sample r% σr % µP − r% σP % SR% logP/D σlogP/D %

1931-2009 0.60 3.00 6.20 19.8 31.3 3.38 45.0

1947-2009 1.00 2.70 6.30 17.6 35.8 3.47 42.9

Model – Baseline calibration

γ ψ β% r% σr % µP − r% σP % SR% logP/D σlogP/D %strip premium

slope

20 years

bond yield %

bond yield

slope

10 1.5 3.5 0.86 2.80 5.06 13.6 37.2 3.51 41.3 < 0 3.56 > 0

Model – No income insurance (φ = 0)

γ ψ β% r% σr % µP − r% σP % SR% logP/D σlogP/D %strip premium

slope

20 years

bond yield %

bond yield

slope

10 1.5 3.5 4.68 1.24 0.24 2.0 11.8 3.54 1.0 ≈ 0 4.48 < 0

Model – Alternative preference settings

γ ψ β% r% σr % µP − r% σP % SR% logP/D σlogP/D %strip premium

slope

20 years

bond yield %

bond yield

slope

10 0.75 3.5 -3.93 5.59 12.90 24.5 52.6 3.23 39.7 > 0 3.71 > 0

1 3.5 -1.21 4.15 8.50 19.0 44.8 3.36 0.0 = 0 3.66 > 0

1.25 3.5 0.07 3.36 6.39 15.8 40.3 3.44 24.6 < 0 3.60 > 0

1.5 3.5 0.86 2.80 5.06 13.6 37.2 3.51 41.3 < 0 3.56 > 0

1.75 3.5 1.37 2.40 4.19 12.0 34.9 3.56 53.5 < 0 3.52 > 0

7.5 0.75 3.5 -2.01 5.59 11.0 24.4 45.0 3.22 39.6 > 0 4.28 > 0

1 3.5 0.15 4.15 7.14 19.0 37.6 3.36 0.0 = 0 4.10 > 0

1.25 3.5 1.14 3.36 5.29 15.8 33.4 3.45 24.6 < 0 3.97 > 0

1.5 3.5 1.74 2.80 4.13 13.6 30.4 3.52 41.4 < 0 3.88 > 0

1.75 3.5 2.11 2.40 3.39 12.0 28.3 3.58 53.6 < 0 3.81 > 0

5 0.75 3.5 -0.05 5.59 9.10 24.4 37.3 3.20 39.5 > 0 4.86 > 0

1 3.5 1.51 4.15 5.78 19.0 30.4 3.36 0.0 = 0 4.54 > 0

1.25 3.5 2.19 3.36 4.21 15.8 26.6 3.46 24.6 < 0 4.34 > 0

1.5 3.5 2.59 2.80 3.23 13.6 23.8 3.54 41.5 < 0 4.19 > 0

1.75 3.5 2.83 2.40 2.61 12.0 21.8 3.60 53.8 < 0 4.09 > 0

72

Table X: Predictability of Excess Returns and Consumption Growth.

The table reports the coefficient, Newey-West t-statistics and adjusted-R2 of the regressions

of cumulative excess returns in Panel A and cumulative consumption growth rates in Panel B

over the horizons of 1, 3, 5 and 7 years on the current log price-dividend ratio:

∑n

i=1∆xt+i = a+ b log P/Dt + εt, ∆xt = {log Rt

Rft−1

, log CtCt−1}.

One thousand simulations are run at the monthly frequency over a 67-year horizon. The

symbols ∗, ∗∗, and ∗∗∗ denote statistical significance at the 10%, 5%, and 1% levels.

Panel A – Excess Returns Panel B – Consumption

Horizon (years) 1 3 5 7 1 3 5 7

log P/Dt -0.98∗∗∗ -2.36∗∗∗ -3.20∗∗∗ -3.68∗∗∗ -0.10∗ -0.27∗ -0.37∗ -0.44∗

t-stat (-3.76) (-4.59) (-6.31) (-7.63) (-2.08) (-1.98) (-2.04) (-2.04)

adj-R2 0.19 0.46 0.60 0.66 0.06 0.15 0.20 0.22

73

Figure 1: The Timing of Macroeconomic Risk and Income Insurance

Left panel: Variance-ratios of value added, wages, EBIT, dividends as a function of

the horizon. Shadow areas denote heteroscedasticity robust 95% confidence intervals.

Right panel: Variance-ratios of value added, wages, EBIT, dividends and value added

minus wages (dashed lines denote heteroscedasticity robust 95% confidence interval) as

a function of the horizon. Data are yearly on the sample 1946-2013.

74

Figure 2: The Term-Structure Effect of Income Insurance

The standardized labor-share (solid line) is plotted together with the standardized 5

years variance ratios of wages minus the variance ratios of dividends (dashed line, left

panel), the 5 years variance ratios of wages (middle), and the 5 years variance ratios of

dividends (right). Data are from the non-financial corporate sector (Flow of Funds) on

the sample 1951:4-2015:1 at quarterly frequency. The time-series of variance ratios are

computed through a rolling window of 10 years.

75

Figure 3: Income Insurance and the Term-Structure of Equity

The standardized labor-share from the non-financial corporate sector (solid line) is

plotted together with the standardized one-year ahead dividend strip return volatility

(dashed line, left panel), the standardized one-year ahead excess return over the risk-free

rate (middle), the standardized one-year ahead excess return over the market return

(right). Data are quarterly on the sample 1996:4-2010:1.

76

Figure 4: Variance Ratios and Persistence

Left panel: Variance-ratios of wages, consumption, and dividends as a function of the

horizon. Dashed lines denote empirical data. Right panel: Autocorrelation function

of the empirical and model-implied labor-share. Cash-flows parameters are from Table

VI.

77

Figure 5: The Term-Structure Effect of Income Insurance

The left and right panels show respectively the variance ratios of dividends and the

difference between the variance ratios of wages and dividends as a function of the

labor-share for several horizons (τ = 2, 5 and 10 years). Cash-flows parameters are

from Table VI.

78

Figure 6: Term-Structures of Dividend Strips

The term-structures of dividend strip return premium, volatility and Sharpe ratio are

reported respectively in the upper left, upper right and lower left panels. The horizontal

dashed line denotes the corresponding moment for equity. The lower right panel shows

the dividend strip return volatility, the equity return volatility, the dividend volatility,

and the consumption volatility. Cash-flows parameters are from Table VI and γ =

10, ψ = 1.5 and β = 3.5%.

79

Figure 7: The Horizon Decomposition of the Equity Premium

Equilibrium density H(t, τ) as a function of the horizon τ (i.e. the relative contribution

of each horizon τ to the equity premium). Solid, dot-dashed and dashed lines denote

respectively the baseline calibration of Table VI (φ = 0.396) and the cases of no income

insurance (φ = 0) and augmented insurance (φ = 3× 0.396).

80

Figure 8: Bond and Equity Yields

Term-structures of equity yields (left upper panel), real bond yields (right upper panel),

expected dividend growth (left lower panel) and premium on equity yields (right lower

panel) at the steady-state as a function of the horizon. Solid and dashed lines denote

respectively the baseline calibration of Table VI and the case of no income insurance

(φ = 0).

81

Figure 9: Business Cycle Uncertainty and Term-Structures

Left upper panel: Term-structures of variance ratios of consumption, wages and div-

idends as a function of the horizon. Dashed lines denote empirical data. Other

panels: Term structures of dividend volatility (right upper panel), dividend strip re-

turn volatility (left lower panel) and dividend strip premium (right lower panel) as

a function of the horizon. Solid lines denote the steady state (zt = z), dashed and

dot-dashed lines denote respectively good and bad states. Cash-flows parameters:

σx = .005, µ = 0.02, λµ = 0.86, σµ = 0.018, z = 0.1, λz = 0.19, σz = 0.026, α = 0.94,

φ = 0.418. Preference parameters: γs = 7.5, ψ = 1.5, β = 4.5%.

82